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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_opt_nlp1_rcomm (e04uf)

Purpose

nag_opt_nlp1_rcomm (e04uf) is designed to minimize an arbitrary smooth function subject to constraints (which may include simple bounds on the variables, linear constraints and smooth nonlinear constraints) using a sequential quadratic programming (SQP) method. You should supply as many first derivatives as possible; any unspecified derivatives are approximated by finite differences. It is not intended for large sparse problems.
nag_opt_nlp1_rcomm (e04uf) may also be used for unconstrained, bound-constrained and linearly constrained optimization.
nag_opt_nlp1_rcomm (e04uf) uses reverse communication for evaluating the objective function, the nonlinear constraint functions and any of their derivatives.

Syntax

[irevcm, iter, istate, c, cjac, clamda, objf, objgrd, r, x, needc, iwork, work, cwsav, lwsav, iwsav, rwsav, ifail] = e04uf(irevcm, a, bl, bu, iter, istate, c, cjac, clamda, objf, objgrd, r, x, iwork, work, cwsav, lwsav, iwsav, rwsav, 'n', n, 'nclin', nclin, 'ncnln', ncnln)
[irevcm, iter, istate, c, cjac, clamda, objf, objgrd, r, x, needc, iwork, work, cwsav, lwsav, iwsav, rwsav, ifail] = nag_opt_nlp1_rcomm(irevcm, a, bl, bu, iter, istate, c, cjac, clamda, objf, objgrd, r, x, iwork, work, cwsav, lwsav, iwsav, rwsav, 'n', n, 'nclin', nclin, 'ncnln', ncnln)
Before calling nag_opt_nlp1_rcomm (e04uf), or either of the option setting functions (e04ue), nag_opt_init (e04wb) must be called.
Note: the interface to this routine has changed since earlier releases of the toolbox:
Mark 24: drop nclin
.

Description

nag_opt_nlp1_rcomm (e04uf) is designed to solve the nonlinear programming problem – the minimization of a smooth nonlinear function subject to a set of constraints on the variables. The problem is assumed to be stated in the following form:
minimize ​ ​F(x)  subject to  l(
x
ALx
c(x)
)
u,
xRn
minimize xRn ​ ​ F(x)   subject to   l x ALx c(x) u,
(1)
where F(x)F(x) (the objective function) is a nonlinear function, ALAL is an nLnL by nn constant matrix, and c(x)c(x) is an nNnN element vector of nonlinear constraint functions. (The matrix ALAL and the vector c(x)c(x) may be empty.) The objective function and the constraint functions are assumed to be smooth, i.e., at least twice-continuously differentiable. (The method of nag_opt_nlp1_rcomm (e04uf) will usually solve (1) if there are only isolated discontinuities away from the solution.)
Note that although the bounds on the variables could be included in the definition of the linear constraints, we prefer to distinguish between them for reasons of computational efficiency. For the same reason, the linear constraints should not be included in the definition of the nonlinear constraints. Upper and lower bounds are specified for all the variables and for all the constraints. An equality constraint can be specified by setting li = uili=ui. If certain bounds are not present, the associated elements of ll or uu can be set to special values that will be treated as - or + +. (See the description of the optional parameter Infinite Bound Size.)
If there are no nonlinear constraints in (1) and FF is linear or quadratic then it will generally be more efficient to use one of nag_opt_lp_solve (e04mf), nag_opt_lsq_lincon_solve (e04nc) or nag_opt_qp_dense_solve (e04nf), or nag_opt_qpconvex2_sparse_solve (e04nq) if the problem is large and sparse. If the problem is large and sparse and does have nonlinear constraints, nag_opt_nlp1_sparse_solve (e04ug) should be used, since nag_opt_nlp1_rcomm (e04uf) treats all matrices as dense.
nag_opt_nlp1_rcomm (e04uf) uses reverse communication for evaluating F(x)F(x), c(x)c(x) and as many of their first partial derivatives as possible; any remaining derivatives are approximated by finite differences. See the description of the optional parameter Derivative Level.
On initial entry, you must supply an initial estimate of the solution to (1).
On intermediate exits, the calling program must compute appropriate values for the objective function, the nonlinear constraints or their derivatives, as specified by the parameter irevcm, and then re-enter the function.
For maximum reliability, it is preferable to provide all partial derivatives (see Chapter 8 of Gill et al. (1981), for a detailed discussion). If they cannot all be provided, it is advisable to provide as many as possible. While developing code to evaluate the objective function and the constraints, the optional parameter Verify should be used to check the calculation of any known derivatives.
The method used by nag_opt_nlp1_rcomm (e04uf) is described in detail in Section [Algorithmic Details].
nag_opt_nlp2_solve (e04wd) is an alternative function which uses a similar method, but with forward communication: that is, the objective and constraint functions are evaluated by functions, supplied as parameters to the function.

References

Dennis J E Jr and Moré J J (1977) Quasi-Newton methods, motivation and theory SIAM Rev. 19 46–89
Dennis J E Jr and Schnabel R B (1981) A new derivation of symmetric positive-definite secant updates nonlinear programming (eds O L Mangasarian, R R Meyer and S M Robinson) 4 167–199 Academic Press
Dennis J E Jr and Schnabel R B (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations Prentice–Hall
Fletcher R (1987) Practical Methods of Optimization (2nd Edition) Wiley
Gill P E, Hammarling S, Murray W, Saunders M A and Wright M H (1986) Users' guide for LSSOL (Version 1.0) Report SOL 86-1 Department of Operations Research, Stanford University
Gill P E, Murray W, Saunders M A and Wright M H (1984a) Procedures for optimization problems with a mixture of bounds and general linear constraints ACM Trans. Math. Software 10 282–298
Gill P E, Murray W, Saunders M A and Wright M H (1984b) Users' guide for SOL/QPSOL version 3.2 Report SOL 84–5 Department of Operations Research, Stanford University
Gill P E, Murray W, Saunders M A and Wright M H (1986a) Some theoretical properties of an augmented Lagrangian merit function Report SOL 86–6R Department of Operations Research, Stanford University
Gill P E, Murray W, Saunders M A and Wright M H (1986b) Users' guide for NPSOL (Version 4.0): a Fortran package for nonlinear programming Report SOL 86-2 Department of Operations Research, Stanford University
Gill P E, Murray W and Wright M H (1981) Practical Optimization Academic Press
Hock W and Schittkowski K (1981) Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems 187 Springer–Verlag
Murtagh B A and Saunders M A (1983) MINOS 5.0 user's guide Report SOL 83-20 Department of Operations Research, Stanford University
Powell M J D (1974) Introduction to constrained optimization Numerical Methods for Constrained Optimization (eds P E Gill and W Murray) 1–28 Academic Press
Powell M J D (1983) Variable metric methods in constrained optimization Mathematical Programming: the State of the Art (eds A Bachem, M Grötschel and B Korte) 288–311 Springer–Verlag

Parameters

Note: this function uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the parameter irevcm. Between intermediate exits and re-entries, all parameters must remain unchanged except those specified by the value of irevcm.

Compulsory Input Parameters

1:     irevcm – int64int32nag_int scalar
On initial entry: must be set to 00.
On intermediate re-entry: must remain unchanged, unless you wish to terminate the solution to the current problem. In this case irevcm may be set to a negative value and then nag_opt_nlp1_rcomm (e04uf) will take a final exit with ifail set to this value of irevcm.
Constraint: irevcm6irevcm6.
2:     a(lda, : :) – double array
The first dimension of the array a must be at least max (1,nclin)max(1,nclin)
The second dimension of the array must be at least nn if nclin > 0nclin>0, and at least 11 otherwise
On initial entry: the iith row of the matrix ALAL of general linear constraints in (1) must be stored in a(i,j)aij, for i = 1,2,,nclini=1,2,,nclin and j = 1,2,,nj=1,2,,n. That is, the iith row contains the coefficients of the iith general linear constraint, for i = 1,2,,nclini=1,2,,nclin.
If nclin = 0nclin=0, the array a is not referenced.
3:     bl(n + nclin + ncnlnn+nclin+ncnln) – double array
4:     bu(n + nclin + ncnlnn+nclin+ncnln) – double array
On initial entry: bl must contain the lower bounds and bu the upper bounds, for all the constraints in the following order. The first nn elements of each array must contain the bounds on the variables, the next nLnL elements the bounds for the general linear constraints (if any) and the next nNnN elements the bounds for the general nonlinear constraints (if any). To specify a nonexistent lower bound (i.e., lj = lj=-), set bl(j)bigbndblj-bigbnd, and to specify a nonexistent upper bound (i.e., uj = + uj=+), set bu(j)bigbndbujbigbnd; the default value of bigbndbigbnd is 10201020, but this may be changed by the optional parameter Infinite Bound Size. To specify the jjth constraint as an equality, set bl(j) = bu(j) = βblj=buj=β, say, where |β| < bigbnd|β|<bigbnd.
Constraints:
  • bl(j)bu(j)bljbuj, for j = 1,2,,n + nclin + ncnlnj=1,2,,n+nclin+ncnln;
  • if bl(j) = bu(j) = βblj=buj=β, |β| < bigbnd|β|<bigbnd.
5:     iter – int64int32nag_int scalar
On intermediate re-entry: must remain unchanged from a previous call to nag_opt_nlp1_rcomm (e04uf).
6:     istate(n + nclin + ncnlnn+nclin+ncnln) – int64int32nag_int array
On initial entry: need not be set if the (default) optional parameter Cold Start is used.
If the optional parameter Warm Start has been chosen, the elements of istate corresponding to the bounds and linear constraints define the initial working set for the procedure that finds a feasible point for the linear constraints and bounds. The active set at the conclusion of this procedure and the elements of istate corresponding to nonlinear constraints then define the initial working set for the first QP subproblem. More precisely, the first nn elements of istate refer to the upper and lower bounds on the variables, the next nLnL elements refer to the upper and lower bounds on ALxALx, and the next nNnN elements refer to the upper and lower bounds on c(x)c(x). Possible values for istate(j)istatej are as follows:
istate(j)istatej Meaning
0 The corresponding constraint is not in the initial QP working set.
1 This inequality constraint should be in the working set at its lower bound.
2 This inequality constraint should be in the working set at its upper bound.
3 This equality constraint should be in the initial working set. This value must not be specified unless bl(j) = bu(j)blj=buj.
The values 2-2, 1-1 and 44 are also acceptable but will be modified by the function. If nag_opt_nlp1_rcomm (e04uf) has been called previously with the same values of n, nclin and ncnln, istate already contains satisfactory information. (See also the description of the optional parameter Warm Start.) The function also adjusts (if necessary) the values supplied in x to be consistent with istate.
Constraint: 2istate(j)4-2istatej4, for j = 1,2,,n + nclin + ncnlnj=1,2,,n+nclin+ncnln.
7:     c( : :) – double array
Note: the dimension of the array c must be at least max (1,ncnln)max(1,ncnln).
On initial entry: need not be set.
On intermediate re-entry: if irevcm = 4irevcm=4 or 66 and needc(i) > 0needci>0, c(i)ci must contain the value of the iith constraint at xx. The remaining elements of c, corresponding to the non-positive elements of needc, are ignored.
8:     cjac(ldcj, : :) – double array
The first dimension of the array cjac must be at least max (1,ncnln)max(1,ncnln)
The second dimension of the array must be at least nn if ncnln > 0ncnln>0, and at least 11 otherwise
On initial entry: in general, cjac need not be initialized before the call to nag_opt_nlp1_rcomm (e04uf). However, if the optional parameter Derivative Level = 2Derivative Level=2 or 33, you may optionally set the constant elements of cjac. Such constant elements need not be re-assigned on subsequent intermediate exits.
If all elements of the constraint Jacobian are known (i.e., Derivative Level = 2Derivative Level=2 or 33), any constant elements may be assigned to cjac one time only at the start of the optimization. An element of cjac that is not subsequently assigned during an intermediate exit will retain its initial value throughout. Constant elements may be loaded into cjac either before the call to nag_opt_nlp1_rcomm (e04uf) or during the first intermediate exit. The ability to preload constants is useful when many Jacobian elements are identically zero, in which case cjac may be initialized to zero and nonzero elements may be reset during intermediate exits.
On intermediate re-entry: if irevcm = 5irevcm=5 or 66 and needc(i) > 0needci>0, the iith row of cjac must contain the available elements of the vector cici given by
ci = ((ci)/(x1),(ci)/(x2),,(ci)/(xn))T,
ci=( ci x1 , ci x2 ,, ci xn )T,
where (ci)/(xj) ci xj  is the partial derivative of the iith constraint with respect to the jjth variable, evaluated at the point xx. The remaining rows of cjac, corresponding to non-positive elements of needc, are ignored. The iith row of the Jacobian should be stored in elements cjac(i,j)cjacij, for i = 1,2,,ncnlni=1,2,,ncnln and j = 1,2,,nj=1,2,,n.
Note that constant nonzero elements do affect the values of the constraints. Thus, if cjac(i,j)cjacij is set to a constant value, it need not be reset during subsequent intermediate exits, but the value cjac(i,j) × x(j)cjacij×xj must nonetheless be added to c(i)ci. For example, if cjac(1,1) = 2cjac11=2 and cjac(1,2) = 5cjac12=-5, then the term 2 × x(1)5 × x(2)2×x1-5×x2 must be included in the definition of c(1)c1.
It must be emphasized that, if Derivative Level = 0Derivative Level=0 or 11, unassigned elements of cjac are not treated as constant; they are estimated by finite differences, at nontrivial expense. If you do not supply a value for the optional parameter Difference Interval, an interval for each element of xx is computed automatically at the start of the optimization. The automatic procedure can usually identify constant elements of cjac, which are then computed once only by finite differences.
See also the description of the optional parameter Verify.
9:     clamda(n + nclin + ncnlnn+nclin+ncnln) – double array
On initial entry: need not be set if the (default) optional parameter Cold Start is used.
If the optional parameter Warm Start has been chosen, clamda(j)clamdaj must contain a multiplier estimate for each nonlinear constraint with a sign that matches the status of the constraint specified by the istate array, for j = n + nclin + 1,,n + nclin + ncnlnj=n+nclin+1,,n+nclin+ncnln. The remaining elements need not be set. Note that if the jjth constraint is defined as ‘inactive’ by the initial value of the istate array (i.e. istate(j) = 0istatej=0), clamda(j)clamdaj should be zero; if the jjth constraint is an inequality active at its lower bound (i.e. istate(j) = 1istatej=1), clamda(j)clamdaj should be non-negative; if the jjth constraint is an inequality active at its upper bound (i.e. istate(j) = 2istatej=2), clamda(j)clamdaj should be non-positive. If necessary, the function will modify clamda to match these rules.
10:   objf – double scalar
On initial entry: need not be set.
On intermediate re-entry: if irevcm = 1irevcm=1 or 33, objf must be set to the value of the objective function at xx.
11:   objgrd(n) – double array
n, the dimension of the array, must satisfy the constraint n > 0n>0.
On initial entry: need not be set.
n, the dimension of the array, must satisfy the constraint n > 0n>0.
On intermediate re-entry: if irevcm = 2irevcm=2 or 33, objgrd must contain the available elements of the gradient evaluated at xx.
See also the description of the optional parameter Verify.
12:   r(ldr,n) – double array
ldr, the first dimension of the array, must satisfy the constraint ldrnldrn.
On initial entry: need not be initialized if the (default) optional parameter Cold Start is used.
If the optional parameter Warm Start has been chosen, r must contain the upper triangular Cholesky factor RR of the initial approximation of the Hessian of the Lagrangian function, with the variables in the natural order. Elements not in the upper triangular part of r are assumed to be zero and need not be assigned.
13:   x(n) – double array
n, the dimension of the array, must satisfy the constraint n > 0n>0.
On initial entry: an initial estimate of the solution.
14:   iwork(liwork) – int64int32nag_int array
liwork, the dimension of the array, must satisfy the constraint liwork3 × n + nclin + 2 × ncnlnliwork3×n+nclin+2×ncnln.
On initial entry: the dimension of the array iwork as declared in the (sub)program from which nag_opt_nlp1_rcomm (e04uf) is called.
Constraint: liwork3 × n + nclin + 2 × ncnlnliwork3×n+nclin+2×ncnln.
15:   work(lwork) – double array
lwork, the dimension of the array, must satisfy the constraint
  • if ncnln = 0ncnln=0 and nclin = 0nclin=0, lwork21 × n + 2lwork21×n+2;
  • if ncnln = 0ncnln=0 and nclin > 0nclin>0, lwork2 × n2 + 21 × n + 11 × nclin + 2lwork2×n2+21×n+11×nclin+2;
  • if ncnln > 0ncnln>0 and nclin0nclin0,
    lwork2 × n2 + n × nclin + 2 × n × ncnln + 21 × n + 11 × nclin +  22 × ncnln + 1lwork2×n2+n×nclin+2×n×ncnln+21×n+11×nclin+22×ncnln+1.
On initial entry: the dimension of the array work as declared in the (sub)program from which nag_opt_nlp1_rcomm (e04uf) is called.
Constraints:
  • if ncnln = 0ncnln=0 and nclin = 0nclin=0, lwork21 × n + 2lwork21×n+2;
  • if ncnln = 0ncnln=0 and nclin > 0nclin>0, lwork2 × n2 + 21 × n + 11 × nclin + 2lwork2×n2+21×n+11×nclin+2;
  • if ncnln > 0ncnln>0 and nclin0nclin0,
    lwork2 × n2 + n × nclin + 2 × n × ncnln + 21 × n + 11 × nclin +  22 × ncnln + 1lwork2×n2+n×nclin+2×n×ncnln+21×n+11×nclin+22×ncnln+1.
The amounts of workspace provided and required may be (by default for nag_opt_nlp1_rcomm (e04uf)) output on the current advisory message unit (as defined by nag_file_set_unit_advisory (x04ab)). As an alternative to computing liwork and lwork from the formulae given above, you may prefer to obtain appropriate values from the output of a preliminary run with liwork and lwork set to 11. (nag_opt_nlp1_rcomm (e04uf) will then terminate with ifail = 9ifail=9.)
16:   cwsav(55) – cell array of strings
17:   lwsav(120120) – logical array
18:   iwsav(610610) – int64int32nag_int array
19:   rwsav(475475) – double array
The arrays lwsav, iwsav, rwsav and cwsav must not be altered between calls to any of the functions nag_opt_init (e04wb), nag_opt_nlp1_rcomm (e04uf), (e04ue).

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the arrays objgrd, x and the first dimension of the array r and the second dimension of the array r. (An error is raised if these dimensions are not equal.)
On initial entry: nn, the number of variables.
Constraint: n > 0n>0.
2:     nclin – int64int32nag_int scalar
Default: The first dimension of the array a.
On initial entry: nLnL, the number of general linear constraints.
Constraint: nclin0nclin0.
3:     ncnln – int64int32nag_int scalar
Default: The first dimension of the array cjac The dimension of the array c.
On initial entry: nNnN, the number of nonlinear constraints.
Constraint: ncnln0ncnln0.

Input Parameters Omitted from the MATLAB Interface

lda ldcj ldr liwork lwork

Output Parameters

1:     irevcm – int64int32nag_int scalar
On intermediate exit: specifies what values the calling program must assign to parameters of nag_opt_nlp1_rcomm (e04uf) before re-entering the function.
irevcm = 1irevcm=1
Set objf to the value of the objective function F(x)F(x).
irevcm = 2irevcm=2
Set objgrd(j)objgrdj to the value (F)/(xj) F xj  if available, for j = 1,2,,nj=1,2,,n.
irevcm = 3irevcm=3
Set objf and objgrd(j)objgrdj as for irevcm = 1irevcm=1 and irevcm = 2irevcm=2.
irevcm = 4irevcm=4
Set c(i)ci to the value of the constraint function ci(x)ci(x), for each ii such that needc(i) > 0needci>0.
irevcm = 5irevcm=5
Set cjac(i,j)cjacij to the value (ci)/(xj) ci xj  if available, for each ii such that needc(i) > 0needci>0 and j = 1,2,,nj=1,2,,n.
irevcm = 6irevcm=6
Set c(i)ci and cjac(i,j)cjacij as for irevcm = 4irevcm=4 and irevcm = 5irevcm=5.
On final exit: irevcm = 0irevcm=0.
2:     iter – int64int32nag_int scalar
On final exit: the number of major iterations performed.
3:     istate(n + nclin + ncnlnn+nclin+ncnln) – int64int32nag_int array
On final exit: the status of the constraints in the QP working set at the point returned in x. The significance of each possible value of istate(j)istatej is as follows:
istate(j)istatej Meaning
2-2 This constraint violates its lower bound by more than the appropriate feasibility tolerance (see the optional parameters Linear Feasibility Tolerance and Nonlinear Feasibility Tolerance). This value can occur only when no feasible point can be found for a QP subproblem.
1-1 This constraint violates its upper bound by more than the appropriate feasibility tolerance (see the optional parameters Linear Feasibility Tolerance and Nonlinear Feasibility Tolerance). This value can occur only when no feasible point can be found for a QP subproblem.
0-0 The constraint is satisfied to within the feasibility tolerance, but is not in the QP working set.
1-1 This inequality constraint is included in the QP working set at its lower bound.
2-2 This inequality constraint is included in the QP working set at its upper bound.
3-3 This constraint is included in the QP working set as an equality. This value of istate can occur only when bl(j) = bu(j)blj=buj.
4:     c( : :) – double array
Note: the dimension of the array c must be at least max (1,ncnln)max(1,ncnln).
On final exit: if ncnln > 0ncnln>0, c(i)ci contains the value of the iith nonlinear constraint function cici at the final iterate, for i = 1,2,,ncnlni=1,2,,ncnln.
If ncnln = 0ncnln=0, the array c is not referenced.
5:     cjac(ldcj, : :) – double array
The first dimension of the array cjac will be max (1,ncnln)max(1,ncnln)
The second dimension of the array will be nn if ncnln > 0ncnln>0, and at least 11 otherwise
ldcjmax (1,ncnln)ldcjmax(1,ncnln).
On final exit: if ncnln > 0ncnln>0, cjac contains the Jacobian matrix of the nonlinear constraint functions at the final iterate, i.e., cjac(i,j)cjacij contains the partial derivative of the iith constraint function with respect to the jjth variable, for i = 1,2,,ncnlni=1,2,,ncnln and j = 1,2,,nj=1,2,,n.
If ncnln = 0ncnln=0, the array cjac is not referenced.
6:     clamda(n + nclin + ncnlnn+nclin+ncnln) – double array
On final exit: the values of the QP multipliers from the last QP subproblem. clamda(j)clamdaj should be non-negative if istate(j) = 1istatej=1 and non-positive if istate(j) = 2istatej=2.
7:     objf – double scalar
On final exit: the value of the objective function at the final iterate.
8:     objgrd(n) – double array
On final exit: the gradient of the objective function at the final iterate (or its finite difference approximation).
9:     r(ldr,n) – double array
ldrnldrn.
On final exit: if Hessian = NOHessian=NO, r contains the upper triangular Cholesky factor RR of QTQQTH~Q, an estimate of the transformed and reordered Hessian of the Lagrangian at xx (see (6) in Section [Overview]).
If Hessian = YESHessian=YES, r contains the upper triangular Cholesky factor RR of HH, the approximate (untransformed) Hessian of the Lagrangian, with the variables in the natural order.
10:   x(n) – double array
On intermediate exit: the point xx at which the objective function, constraint functions or their derivatives are to be evaluated.
On final exit: the final estimate of the solution.
11:   needc(max (1,ncnln)max(1,ncnln)) – int64int32nag_int array
On intermediate exit: if irevcm4irevcm4, needc specifies the indices of the elements of c and/or cjac that must be assigned. If needc(i) > 0needci>0, then the iith element of c and/or the available elements of the iith row of cjac must be evaluated at xx.
12:   iwork(liwork) – int64int32nag_int array
13:   work(lwork) – double array
The amounts of workspace provided and required may be (by default for nag_opt_nlp1_rcomm (e04uf)) output on the current advisory message unit (as defined by nag_file_set_unit_advisory (x04ab)). As an alternative to computing liwork and lwork from the formulae given above, you may prefer to obtain appropriate values from the output of a preliminary run with liwork and lwork set to 11. (nag_opt_nlp1_rcomm (e04uf) will then terminate with ifail = 9ifail=9.)
14:   cwsav(55) – cell array of strings
15:   lwsav(120120) – logical array
16:   iwsav(610610) – int64int32nag_int array
17:   rwsav(475475) – double array
18:   ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).
nag_opt_nlp1_rcomm (e04uf) returns with ifail = 0ifail=0 if the iterates have converged to a point xx that satisfies the first-order Kuhn–Tucker conditions (see Section [Overview]) to the accuracy requested by the optional parameter Optimality Tolerance. This has default value = εr0.8default value=εr0.8, where εrεr is the value of the optional parameter Function Precision (default value = ε0.9default value=ε0.9, where εε is the machine precision). That is ifail = 0ifail=0 when the projected gradient and active constraint residuals are negligible at xx.
You should check whether the following four conditions are satisfied:
(i) the final value of Norm Gz (see Section [Printed output]) is significantly less than that at the starting point;
(ii) during the final major iterations, the values of Step and Mnr (see Section [Printed output]) are both one;
(iii) the last few values of both Norm Gz and Violtn (see Section [Printed output]) become small at a fast linear rate; and
(iv) Cond Hz (see Section [Printed output]) is small.
If all these conditions hold, xx is almost certainly a local minimum of (1).

Error Indicators and Warnings

Note: nag_opt_nlp1_rcomm (e04uf) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W ifail < 0ifail<0
A negative value of ifail indicates an exit from nag_opt_nlp1_rcomm (e04uf) because you set irevcm < 0irevcm<0 during an intermediate exit. The value of ifail will be the same as your setting of irevcm.
W ifail = 1ifail=1
The final iterate xx satisfies the first-order Kuhn–Tucker conditions (see Section [Overview]) to the accuracy requested, but the sequence of iterates has not yet converged. nag_opt_nlp1_rcomm (e04uf) was terminated because no further improvement could be made in the merit function (see Section [Printed output]).
This value of ifail may occur in several circumstances. The most common situation is that you ask for a solution with accuracy that is not attainable with the given precision of the problem (as specified by the optional parameter Function Precision). This condition will also occur if, by chance, an iterate is an ‘exact’ Kuhn–Tucker point, but the change in the variables was significant at the previous iteration. (This situation often happens when minimizing very simple functions, such as quadratics.)
If the four conditions listed in Section [Parameters] for ifail = 0ifail=0 are satisfied, xx is likely to be a solution of (1) even if ifail = 1ifail=1.
W ifail = 2ifail=2
nag_opt_nlp1_rcomm (e04uf) has terminated without finding a feasible point for the linear constraints and bounds, which means that either no feasible point exists for the given value of the optional parameter Linear Feasibility Tolerance, or no feasible point could be found in the number of iterations specified by the optional parameter Minor Iteration Limit. You should check that there are no constraint redundancies. If the data for the constraints are accurate only to an absolute precision σσ, you should ensure that the value of the optional parameter Linear Feasibility Tolerance is greater than σσ. For example, if all elements of ALAL are of order unity and are accurate to only three decimal places, Linear Feasibility Tolerance should be at least 10310-3.
W ifail = 3ifail=3
No feasible point could be found for the nonlinear constraints. The problem may have no feasible solution. This means that there has been a sequence of QP subproblems for which no feasible point could be found (indicated by I at the end of each line of intermediate printout produced by the major iterations; see Section [Printed output]). This behaviour will occur if there is no feasible point for the nonlinear constraints. (However, there is no general test that can determine whether a feasible point exists for a set of nonlinear constraints.) If the infeasible subproblems occur from the very first major iteration, it is highly likely that no feasible point exists. If infeasibilities occur when earlier subproblems have been feasible, small constraint inconsistencies may be present. You should check the validity of constraints with negative values of istate. If you are convinced that a feasible point does exist, nag_opt_nlp1_rcomm (e04uf) should be restarted at a different starting point.
  ifail = 4ifail=4
The limiting number of iterations (as determined by the optional parameter Major Iteration Limit) has been reached.
If the algorithm appears to be making satisfactory progress, then optional parameter Major Iteration Limit may be too small. If so, either increase its value and rerun nag_opt_nlp1_rcomm (e04uf) or, alternatively, rerun nag_opt_nlp1_rcomm (e04uf) using the optional parameter Warm Start. If the algorithm seems to be making little or no progress however, then you should check for incorrect gradients or ill-conditioning as described under ifail = 6ifail=6.
Note that ill-conditioning in the working set is sometimes resolved automatically by the algorithm, in which case performing additional iterations may be helpful. However, ill-conditioning in the Hessian approximation tends to persist once it has begun, so that allowing additional iterations without altering rr is usually inadvisable. If the quasi-Newton update of the Hessian approximation was reset during the latter major iterations (i.e., an R occurs at the end of each line of intermediate printout; see Section [Printed output]), it may be worthwhile to try a Warm Start at the final point as suggested above.
  ifail = 5ifail=5
Not used by this function.
W ifail = 6ifail=6
xx does not satisfy the first-order Kuhn–Tucker conditions (see Section [Overview]), and no improved point for the merit function (see Section [Printed output]) could be found during the final linesearch.
This sometimes occurs because an overly stringent accuracy has been requested, i.e., the value of the optional parameter Optimality Tolerance (default value = εr0.8default value=εr0.8, where εrεr is the value of the optional parameter Function Precision) is too small. In this case you should apply the four tests described under ifail = 0ifail=0 to determine whether or not the final solution is acceptable (see Gill et al. (1981), for a discussion of the attainable accuracy).
If many iterations have occurred in which essentially no progress has been made and nag_opt_nlp1_rcomm (e04uf) has failed completely to move from the initial point, then values set by the calling program for the objective or constraint functions or their derivatives during intermediate exits may be incorrect. You should refer to comments under ifail = 7ifail=7 and check the gradients using the optional parameter Verify. Unfortunately, there may be small errors in the objective and constraint gradients that cannot be detected by the verification process. Finite difference approximations to first derivatives are catastrophically affected by even small inaccuracies. An indication of this situation is a dramatic alteration in the iterates if the finite difference interval is altered. One might also suspect this type of error if a switch is made to central differences even when Norm Gz and Violtn (see Section [Printed output]) are large.
Another possibility is that the search direction has become inaccurate because of ill-conditioning in the Hessian approximation or the matrix of constraints in the working set; either form of ill-conditioning tends to be reflected in large values of Mnr (the number of iterations required to solve each QP subproblem; see Section [Printed output]).
If the condition estimate of the projected Hessian (Cond Hz; see Section [Printed output]) is extremely large, it may be worthwhile rerunning nag_opt_nlp1_rcomm (e04uf) from the final point with the optional parameter Warm Start. In this situation, istate and clamda should be left unaltered and rr should be reset to the identity matrix.
If the matrix of constraints in the working set is ill-conditioned (i.e., Cond T is extremely large; see Section [Description of Monitoring Information]), it may be helpful to run nag_opt_nlp1_rcomm (e04uf) with a relaxed value of the optional parameter Feasibility Tolerance. (Constraint dependencies are often indicated by wide variations in size in the diagonal elements of the matrix TT, whose diagonals will be printed if Major Print Level30Major Print Level30.)
  ifail = 7ifail=7
The user-supplied derivatives of the objective function and/or nonlinear constraints appear to be incorrect.
Large errors were found in the derivatives of the objective function and/or nonlinear constraints. This value of ifail will occur if the verification process indicated that at least one gradient or Jacobian element had no correct figures. You should refer to the printed output to determine which elements are suspected to be in error.
As a first-step, you should check that the code for the objective and constraint values is correct – for example, by computing the function at a point where the correct value is known. However, care should be taken that the chosen point fully tests the evaluation of the function. It is remarkable how often the values x = 0x=0 or x = 1x=1 are used in such a test, and how often the special properties of these numbers make the test meaningless.
Special care should be used in the test if computation of the objective function involves subsidiary data communicated in global storage. Although the first evaluation of the function may be correct, subsequent calculations may be in error because some of the subsidiary data has accidentally been overwritten.
Gradient checking will be ineffective if the objective function uses information computed by the constraints, since they are not necessarily computed before each function evaluation.
Errors in programming the function may be quite subtle in that the function value is ‘almost’ correct. For example, the function may not be accurate to full precision because of the inaccurate calculation of a subsidiary quantity, or the limited accuracy of data upon which the function depends. A common error on machines where numerical calculations are usually performed in double precision is to include even one single precision constant in the calculation of the function; since some compilers do not convert such constants to double precision, half the correct figures may be lost by such a seemingly trivial error.
  ifail = 8ifail=8
Not used by this function.
  ifail = 9ifail=9
An input parameter is invalid.
  OverflowOverflow
If overflow occurs then either an element of CC is very large, or the singular values or singular vectors have been incorrectly supplied.

Accuracy

If ifail = 0ifail=0 on final exit then the vector returned in the array x is an estimate of the solution to an accuracy of approximately Optimality Tolerance (default value = ε0.8default value=ε0.8, where εε is the machine precision).

Further Comments

Description of the Printed Output

This section describes the intermediate printout and final printout produced by nag_opt_nlp1_rcomm (e04uf). The intermediate printout is a subset of the monitoring information produced by nag_opt_nlp1_rcomm (e04uf) at every iteration (see Section [Description of Monitoring Information]). You can control the level of printed output (see the description of the optional parameter Major Print Level). Note that the intermediate printout and final printout are produced only if Major Print Level10Major Print Level10 (the default for nag_opt_nlp1_rcomm (e04uf), by default no output is produced by nag_opt_nlp1_rcomm (e04uf)).
The following line of summary output ( < 80<80 characters) is produced at every major iteration. In all cases, the values of the quantities printed are those in effect on completion of the given iteration.
Maj is the major iteration count.
Mnr is the number of minor iterations required by the feasibility and optimality phases of the QP subproblem. Generally, Mnr will be 11 in the later iterations, since theoretical analysis predicts that the correct active set will be identified near the solution (see Section [Algorithmic Details]).
Note that Mnr may be greater than the optional parameter Minor Iteration Limit if some iterations are required for the feasibility phase.
Step is the step αkαk taken along the computed search direction. On reasonably well-behaved problems, the unit step (i.e., αk = 1αk=1) will be taken as the solution is approached.
Merit Function is the value of the augmented Lagrangian merit function (12) at the current iterate. This function will decrease at each iteration unless it was necessary to increase the penalty parameters (see Section [The Merit Function]). As the solution is approached, Merit Function will converge to the value of the objective function at the solution.
If the QP subproblem does not have a feasible point (signified by I at the end of the current output line) then the merit function is a large multiple of the constraint violations, weighted by the penalty parameters. During a sequence of major iterations with infeasible subproblems, the sequence of Merit Function values will decrease monotonically until either a feasible subproblem is obtained or nag_opt_nlp1_rcomm (e04uf) terminates with ifail = 3ifail=3 (no feasible point could be found for the nonlinear constraints).
If there are no nonlinear constraints present (i.e., ncnln = 0ncnln=0) then this entry contains Objective, the value of the objective function F(x)F(x). The objective function will decrease monotonically to its optimal value when there are no nonlinear constraints.
Norm Gz is ZTgFRZTgFR, the Euclidean norm of the projected gradient (see Section [Solution of the Quadratic Programming Subproblem]). Norm Gz will be approximately zero in the neighbourhood of a solution.
Violtn is the Euclidean norm of the residuals of constraints that are violated or in the predicted active set (not printed if ncnln is zero). Violtn will be approximately zero in the neighbourhood of a solution.
Cond Hz is a lower bound on the condition number of the projected Hessian approximation HZ HZ  ( HZ = ZT HFR Z = RZT RZ HZ = ZT HFR Z = RZT RZ ; see (6)). The larger this number, the more difficult the problem.
M is printed if the quasi-Newton update has been modified to ensure that the Hessian approximation is positive definite (see Section [The Quasi-Newton Update]).
I is printed if the QP subproblem has no feasible point.
C is printed if central differences have been used to compute the unspecified objective and constraint gradients. If the value of Step is zero then the switch to central differences was made because no lower point could be found in the linesearch. (In this case, the QP subproblem is resolved with the central difference gradient and Jacobian.) If the value of Step is nonzero then central differences were computed because Norm Gz and Violtn imply that xx is close to a Kuhn–Tucker point (see Section [Overview]).
L is printed if the linesearch has produced a relative change in xx greater than the value defined by the optional parameter Step Limit. If this output occurs frequently during later iterations of the run, optional parameter Step Limit should be set to a larger value.
R is printed if the approximate Hessian has been refactorized. If the diagonal condition estimator of RR indicates that the approximate Hessian is badly conditioned then the approximate Hessian is refactorized using column interchanges. If necessary, RR is modified so that its diagonal condition estimator is bounded.
The final printout includes a listing of the status of every variable and constraint.
The following describes the printout for each variable. A full stop (.) is printed for any numerical value that is zero.
Varbl gives the name (V) and index jj, for j = 1,2,,nj=1,2,,n, of the variable.
State gives the state of the variable (FR if neither bound is in the working set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound, TF if temporarily fixed at its current value). If Value lies outside the upper or lower bounds by more than the Feasibility Tolerance, State will be ++ or -- respectively.
A key is sometimes printed before State.
A Alternative optimum possible. The variable is active at one of its bounds, but its Lagrange multiplier is essentially zero. This means that if the variable were allowed to start moving away from its bound then there would be no change to the objective function. The values of the other free variables might change, giving a genuine alternative solution. However, if there are any degenerate variables (labelled D), the actual change might prove to be zero, since one of them could encounter a bound immediately. In either case the values of the Lagrange multipliers might also change.
D Degenerate. The variable is free, but it is equal to (or very close to) one of its bounds.
I Infeasible. The variable is currently violating one of its bounds by more than the Feasibility Tolerance.
Value is the value of the variable at the final iteration.
Lower Bound is the lower bound specified for the variable. None indicates that bl(j)bigbndblj-bigbnd.
Upper Bound is the upper bound specified for the variable. None indicates that bu(j)bigbndbujbigbnd.
Lagr Mult is the Lagrange multiplier for the associated bound. This will be zero if State is FR unless bl(j)bigbndblj-bigbnd and bu(j)bigbndbujbigbnd, in which case the entry will be blank. If xx is optimal, the multiplier should be non-negative if State is LL and non-positive if State is UL.
Slack is the difference between the variable Value and the nearer of its (finite) bounds bl(j)blj and bu(j)buj. A blank entry indicates that the associated variable is not bounded (i.e., bl(j)bigbndblj-bigbnd and bu(j)bigbndbujbigbnd).
The meaning of the printout for linear and nonlinear constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, bl(j)blj and bu(j)buj replaced by bl(n + j)bln+j and bu(n + j)bun+j respectively and with the following changes in the heading:
L Con gives the name (L) and index jj, for j = 1,2,,nLj=1,2,,nL, of the linear constraint.
N Con gives the name (N) and index (jnLj-nL), for j = nL + 1,,nL + nNj=nL+1,,nL+nN, of the nonlinear constraint.
Note that movement off a constraint (as opposed to a variable moving away from its bound) can be interpreted as allowing the entry in the Slack column to become positive.
Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision.

Example

function nag_opt_nlp1_rcomm_example
irevcm = int64(0);
nclin = 1;
a = [1, 1, 1, 1];
bl = [1;
     1;
     1;
     1;
     -1e25;
     -1e25;
     25];
bu = [5;
     5;
     5;
     5;
     20;
     40;
     1e25];
iter = int64(0);
istate = zeros(7, 1, 'int64');
c = [0;
     0];
cjac = [0, 0, 0, 0;
     0, 0, 0, 0];
clamda = zeros(7, 1);
objf = 0;
objgrd = zeros(4, 1);
r = zeros(4, 4);
x = [1; 5; 5; 1];
iwork = zeros(100, 1, 'int64');
work = zeros(1000, 1);
[cwsav,lwsav,iwsav,rwsav,ifail] = nag_opt_init('nag_opt_nlp1_rcomm');
if (ifail == 0)
  [irevcm, iter, istate, c, cjac, clamda, objf, objgrd, r, x, needc, ...
   iwork, work, cwsav, lwsav, iwsav, rwsav, ifail] = ...
     nag_opt_nlp1_rcomm(irevcm, a, bl, bu, iter, istate, c, cjac, clamda, ...
            objf, objgrd, r, x, iwork, work, cwsav, lwsav, iwsav, rwsav);

  while (irevcm > 0)

    if (irevcm == 1 || irevcm == 3)
      % Evaluate the objective function.
      objf = x(1)*x(4)*(x(1)+x(2)+x(3)) + x(3);
    end

    if (irevcm == 2  ||  irevcm == 3)
      % Evaluate the objective gradient.
      objgrd(1) = x(4)*(2*x(1)+x(2)+x(3));
      objgrd(2) = x(1)*x(4);
      objgrd(3) = x(1)*x(4) + 1;
      objgrd(4) = x(1)*(x(1)+x(2)+x(3));
    end

    if (irevcm == 4  ||  irevcm == 6)
      % Evaluate the nonlinear constraint functions.
      if (needc(1) > 0)
        c(1) = x(1)^2 + x(2)^2 + x(3)^2 + x(4)^2;
      end
      if (needc(2) > 0)
        c(2) = x(1)*x(2)*x(3)*x(4);
      end
    end

    if (irevcm == 5  ||  irevcm == 6)
      % Evaluate the constraint Jacobian.
      if (needc(1) > 0)
        cjac(1,1) = 2*x(1);
        cjac(1,2) = 2*x(2);
        cjac(1,3) = 2*x(3);
        cjac(1,4) = 2*x(4);
      end
      if (needc(2) > 0)
        cjac(2,1) = x(2)*x(3)*x(4);
        cjac(2,2) = x(1)*x(3)*x(4);
        cjac(2,3) = x(1)*x(2)*x(4);
        cjac(2,4) = x(1)*x(2)*x(3);
      end
    end

    [irevcm, iter, istate, c, cjac, clamda, objf, objgrd, r, x, needc, ...
     iwork, work, cwsav, lwsav, iwsav, rwsav, ifail] = ...
       nag_opt_nlp1_rcomm(irevcm, a, bl, bu, iter, istate, c, cjac, clamda, ...
              objf, objgrd, r, x, iwork, work, cwsav, lwsav, iwsav, rwsav);

  end

  if (ifail == 0)
    fprintf('\n Varbl  Istate   Value         Lagr Mult\n');
    for i=1:4
      fprintf(' V %3d %3d %14.4f %12.4f \n', i, istate(i), x(i), clamda(i));
    end
    ax = a*x;
    fprintf('\n L Con  Istate   Value         Lagr Mult\n')
    for i=5:4+nclin
      j=i-4;
      fprintf(' L %3d %3d  %14.4f%12.4f\n', j, istate(i), ax(j), clamda(i));
    end
    fprintf('\n L Con  Istate   Value         Lagr Mult\n')
    for i=5+nclin:6+nclin
      j=i-4-nclin;
      fprintf(' N %3d %3d  %14.4f%12.4f\n', j, istate(i), c(j), clamda(i));
    end
    fprintf('\n Final objective value = %15.7f\n', objf)
  end

end
 

 Varbl  Istate   Value         Lagr Mult
 V   1   1         1.0000       1.0879 
 V   2   0         4.7430       0.0000 
 V   3   0         3.8211       0.0000 
 V   4   0         1.3794       0.0000 

 L Con  Istate   Value         Lagr Mult
 L   1   0         10.9436      0.0000

 L Con  Istate   Value         Lagr Mult
 N   1   2         40.0000     -0.1615
 N   2   1         25.0000      0.5523

 Final objective value =      17.0140173

function e04uf_example
nclin = 1;
irevcm = int64(0);
a = [1, 1, 1, 1];
bl = [1;
     1;
     1;
     1;
     -1e25;
     -1e25;
     25];
bu = [5;
     5;
     5;
     5;
     20;
     40;
     1e25];
iter = int64(0);
istate = zeros(7, 1, 'int64');
c = [0;
     0];
cjac = [0, 0, 0, 0;
     0, 0, 0, 0];
clamda = zeros(7, 1);
objf = 0;
objgrd = zeros(4, 1);
r = zeros(4, 4);
x = [1; 5; 5; 1];
iwork = zeros(100, 1, 'int64');
work = zeros(1000, 1);
[cwsav,lwsav,iwsav,rwsav,ifail] = e04wb('e04uf');
if (ifail == 0)
  [irevcm, iter, istate, c, cjac, clamda, objf, objgrd, r, x, needc, ...
   iwork, work, cwsav, lwsav, iwsav, rwsav, ifail] = ...
     e04uf(irevcm, a, bl, bu, iter, istate, c, cjac, clamda, ...
            objf, objgrd, r, x, iwork, work, cwsav, lwsav, iwsav, rwsav);

  while (irevcm > 0)

    if (irevcm == 1 || irevcm == 3)
      % Evaluate the objective function.
      objf = x(1)*x(4)*(x(1)+x(2)+x(3)) + x(3);
    end

    if (irevcm == 2  ||  irevcm == 3)
      % Evaluate the objective gradient.
      objgrd(1) = x(4)*(2*x(1)+x(2)+x(3));
      objgrd(2) = x(1)*x(4);
      objgrd(3) = x(1)*x(4) + 1;
      objgrd(4) = x(1)*(x(1)+x(2)+x(3));
    end

    if (irevcm == 4  ||  irevcm == 6)
      % Evaluate the nonlinear constraint functions.
      if (needc(1) > 0)
        c(1) = x(1)^2 + x(2)^2 + x(3)^2 + x(4)^2;
      end
      if (needc(2) > 0)
        c(2) = x(1)*x(2)*x(3)*x(4);
      end
    end

    if (irevcm == 5  ||  irevcm == 6)
      % Evaluate the constraint Jacobian.
      if (needc(1) > 0)
        cjac(1,1) = 2*x(1);
        cjac(1,2) = 2*x(2);
        cjac(1,3) = 2*x(3);
        cjac(1,4) = 2*x(4);
      end
      if (needc(2) > 0)
        cjac(2,1) = x(2)*x(3)*x(4);
        cjac(2,2) = x(1)*x(3)*x(4);
        cjac(2,3) = x(1)*x(2)*x(4);
        cjac(2,4) = x(1)*x(2)*x(3);
      end
    end

    [irevcm, iter, istate, c, cjac, clamda, objf, objgrd, r, x, needc, ...
     iwork, work, cwsav, lwsav, iwsav, rwsav, ifail] = ...
       e04uf(irevcm, a, bl, bu, iter, istate, c, cjac, clamda, ...
              objf, objgrd, r, x, iwork, work, cwsav, lwsav, iwsav, rwsav);

  end

  if (ifail == 0)
    fprintf('\n Varbl  Istate   Value         Lagr Mult\n');
    for i=1:4
      fprintf(' V %3d %3d %14.4f %12.4f \n', i, istate(i), x(i), clamda(i));
    end
    ax = a*x;
    fprintf('\n L Con  Istate   Value         Lagr Mult\n')
    for i=5:4+nclin
      j=i-4;
      fprintf(' L %3d %3d  %14.4f%12.4f\n', j, istate(i), ax(j), clamda(i));
    end
    fprintf('\n L Con  Istate   Value         Lagr Mult\n')
    for i=5+nclin:6+nclin
      j=i-4-nclin;
      fprintf(' N %3d %3d  %14.4f%12.4f\n', j, istate(i), c(j), clamda(i));
    end
    fprintf('\n Final objective value = %15.7f\n', objf)
  end

end
 

 Varbl  Istate   Value         Lagr Mult
 V   1   1         1.0000       1.0879 
 V   2   0         4.7430       0.0000 
 V   3   0         3.8211       0.0000 
 V   4   0         1.3794       0.0000 

 L Con  Istate   Value         Lagr Mult
 L   1   0         10.9436      0.0000

 L Con  Istate   Value         Lagr Mult
 N   1   2         40.0000     -0.1615
 N   2   1         25.0000      0.5523

 Final objective value =      17.0140173

the remainder of this document is intended for more advanced users. Section [Algorithmic Details] contains a detailed description of the algorithm which may be needed in order to understand Sections [Optional Parameters] and [Description of Monitoring Information]. Section [Optional Parameters] describes the optional parameters which may be set by calls to nag_opt_nlp1_option_string (e04ue). Section [Description of Monitoring Information] describes the quantities which can be requested to monitor the course of the computation.

Algorithmic Details

This section contains a detailed description of the method used by nag_opt_nlp1_rcomm (e04uf).

Overview

nag_opt_nlp1_rcomm (e04uf) is essentially identical to the function NPSOL described in Gill et al. (1986b).
At a solution of (1), some of the constraints will be active, i.e., satisfied exactly. An active simple bound constraint implies that the corresponding variable is fixed at its bound, and hence the variables are partitioned into fixed and free variables. Let CC denote the mm by nn matrix of gradients of the active general linear and nonlinear constraints. The number of fixed variables will be denoted by nFX nFX , with nFR nFR  ( nFR = nnFX nFR = n-nFX ) the number of free variables. The subscripts ‘FX’ and ‘FR’ on a vector or matrix will denote the vector or matrix composed of the elements corresponding to fixed or free variables.
A point xx is a first-order Kuhn–Tucker point for (1) (see Powell (1974)) if the following conditions hold:
(i) xx is feasible;
(ii) there exist vectors ξξ and λλ (the Lagrange multiplier vectors for the bound and general constraints) such that
g = CTλ + ξ
g=CTλ+ξ
(2)
where gg is the gradient of FF evaluated at xx and ξj = 0ξj=0 if the jjth variable is free.
(iii) the Lagrange multiplier corresponding to an inequality constraint active at its lower bound must be non-negative. It is non-positive for an inequality constraint active at its upper bound.
Let ZZ denote a matrix whose columns form a basis for the set of vectors orthogonal to the rows of CFRCFR; i.e., CFRZ = 0CFRZ=0. An equivalent statement of the condition (2) in terms of ZZ is
ZTgFR = 0.
ZTgFR=0.
The vector ZTgFRZTgFR is termed the projected gradient of FF at xx. Certain additional conditions must be satisfied in order for a first-order Kuhn–Tucker point to be a solution of (1) (see Powell (1974)).
nag_opt_nlp1_rcomm (e04uf) implements a sequential quadratic programming (SQP) method. For an overview of SQP methods, see Fletcher (1987), Gill et al. (1981) and Powell (1983).
The basic structure of nag_opt_nlp1_rcomm (e04uf) involves major and minor iterations. The major iterations generate a sequence of iterates {xk}{xk} that converge to x*x*, a first-order Kuhn–Tucker point of (1). At a typical major iteration, the new iterate xx- is defined by
x = x + αp
x-=x+αp
(3)
where xx is the current iterate, the non-negative scalar αα is the step length, and pp is the search direction. (For simplicity, we shall always consider a typical iteration and avoid reference to the index of the iteration.) Also associated with each major iteration are estimates of the Lagrange multipliers and a prediction of the active set.
The search direction pp in (3) is the solution of a quadratic programming subproblem of the form
minimize gTp + (1/2)pTHp  subject to  l{
p
ALp
ANp
}
u,
p
minimizep gT p+ 12 pT H p   subject to   l- { p ALp ANp } u- ,
(4)
where gg is the gradient of FF at xx, the matrix HH is a positive definite quasi-Newton approximation to the Hessian of the Lagrangian function (see Section [The Quasi-Newton Update]), and ANAN is the Jacobian matrix of cc evaluated at xx. (Finite difference estimates may be used for gg and ANAN; see the optional parameter Derivative Level.) Let ll in (1) be partitioned into three sections: lBlB, lLlL and lNlN, corresponding to the bound, linear and nonlinear constraints. The vector ll- in (4) is similarly partitioned and is defined as
lB = lBx,  lL = lLALx,   and  lN = lNc,
l-B=lB-x,  l-L=lL-ALx,   and  l-N=lN-c,
where cc is the vector of nonlinear constraints evaluated at xx. The vector uu- is defined in an analogous fashion.
The estimated Lagrange multipliers at each major iteration are the Lagrange multipliers from the subproblem (4) (and similarly for the predicted active set). (The numbers of bounds, general linear and nonlinear constraints in the QP active set are the quantities Bnd, Lin and Nln in the monitoring file output of nag_opt_nlp1_rcomm (e04uf); see Section [Description of Monitoring Information].) In nag_opt_nlp1_rcomm (e04uf), (4) is solved using nag_opt_lsq_lincon_solve (e04nc). Since solving a quadratic program is itself an iterative procedure, the minor iterations of nag_opt_nlp1_rcomm (e04uf) are the iterations of nag_opt_lsq_lincon_solve (e04nc). (More details about solving the subproblem are given in Section [Solution of the Quadratic Programming Subproblem].)
Certain matrices associated with the QP subproblem are relevant in the major iterations. Let the subscripts ‘FX’ and ‘FR’ refer to the predicted fixed and free variables, and let CC denote the mm by nn matrix of gradients of the general linear and nonlinear constraints in the predicted active set. Firstly, we have available the TQTQ factorization of CFRCFR:
CFRQFR = (0T),
CFRQFR=(0T),
(5)
where TT is a nonsingular mm by mm reverse-triangular matrix (i.e., tij = 0tij=0 if i + j < mi+j<m), and the nonsingular nFRnFR by nFRnFR matrix QFRQFR is the product of orthogonal transformations (see Gill et al. (1984b)). Secondly, we have the upper triangular Cholesky factor RR of the transformed and reordered Hessian matrix
RTR = HQQTQ,
RTR=HQQTH~Q,
(6)
where H~ is the Hessian HH with rows and columns permuted so that the free variables are first and QQ is the nn by nn matrix
Q =
(QFR)
IFX
Q= QFR IFX
(7)
with IFXIFX the identity matrix of order nFXnFX. If the columns of QFRQFR are partitioned so that
QFR =
(ZY)
,
QFR= Z Y ,
then the nZnZ (nZnFRmnZnFR-m) columns of ZZ form a basis for the null space of CFRCFR. The matrix ZZ is used to compute the projected gradient ZTgFRZTgFR at the current iterate. (The values Nz and Norm Gz printed by nag_opt_nlp1_rcomm (e04uf) give nZnZ and ZTgFRZTgFR, see Section [Description of Monitoring Information].)
A theoretical characteristic of SQP methods is that the predicted active set from the QP subproblem (4) is identical to the correct active set in a neighbourhood of x*x*. In nag_opt_nlp1_rcomm (e04uf), this feature is exploited by using the QP active set from the previous iteration as a prediction of the active set for the next QP subproblem, which leads in practice to optimality of the subproblems in only one iteration as the solution is approached. Separate treatment of bound and linear constraints in nag_opt_nlp1_rcomm (e04uf) also saves computation in factorizing CFRCFR and HQHQ.
Once pp has been computed, the major iteration proceeds by determining a step length αα that produces a ‘sufficient decrease’ in an augmented Lagrangian merit function (see Section [The Merit Function]). Finally, the approximation to the transformed Hessian matrix HQHQ is updated using a modified BFGS quasi-Newton update (see Section [The Quasi-Newton Update]) to incorporate new curvature information obtained in the move from xx to xx-.
On entry to nag_opt_nlp1_rcomm (e04uf), an iterative procedure from nag_opt_lsq_lincon_solve (e04nc) is executed, starting with the user-supplied initial point, to find a point that is feasible with respect to the bounds and linear constraints (using the tolerance specified by the optional parameter Linear Feasibility Tolerance). If no feasible point exists for the bound and linear constraints, (1) has no solution and nag_opt_nlp1_rcomm (e04uf) terminates. Otherwise, the problem functions will thereafter be evaluated only at points that are feasible with respect to the bounds and linear constraints. The only exception involves variables whose bounds differ by an amount comparable to the finite difference interval (see the discussion of the optional parameter Difference Interval). In contrast to the bounds and linear constraints, it must be emphasized that the nonlinear constraints will not generally be satisfied until an optimal point is reached.
Facilities are provided to check whether the user-supplied gradients appear to be correct (see the description of the optional parameter Verify). In general, the check is provided at the first point that is feasible with respect to the linear constraints and bounds. However, you may request that the check be performed at the initial point.
In summary, the method of nag_opt_nlp1_rcomm (e04uf) first determines a point that satisfies the bound and linear constraints. Thereafter, each iteration includes:
(a) the solution of a quadratic programming subproblem;
(b) a linesearch with an augmented Lagrangian merit function; and
(c) a quasi-Newton update of the approximate Hessian of the Lagrangian function.
These three procedures are described in more detail in Sections [Solution of the Quadratic Programming Subproblem] to [The Quasi-Newton Update].

Solution of the Quadratic Programming Subproblem

The search direction pp is obtained by solving (4) using nag_opt_lsq_lincon_solve (e04nc) (see Gill et al. (1986)), which was specifically designed to be used within an SQP algorithm for nonlinear programming.
nag_opt_lsq_lincon_solve (e04nc) is based on a two-phase (primal) quadratic programming method. The two phases of the method are: finding an initial feasible point by minimizing the sum of infeasibilities (the feasibility phase) and minimizing the quadratic objective function within the feasible region (the optimality phase). The computations in both phases are performed by the same functions. The two-phase nature of the algorithm is reflected by changing the function being minimized from the sum of infeasibilities to the quadratic objective function.
In general, a quadratic program must be solved by iteration. Let pp denote the current estimate of the solution of (4); the new iterate pp- is defined by
p = p + σd
p-=p+σd
(8)
where, as in (3), σσ is a non-negative step length and dd is a search direction.
At the beginning of each iteration of nag_opt_lsq_lincon_solve (e04nc), a working set is defined of constraints (general and bound) that are satisfied exactly. The vector dd is then constructed so that the values of constraints in the working set remain unaltered for any move along dd. For a bound constraint in the working set, this property is achieved by setting the corresponding element of dd to zero, i.e., by fixing the variable at its bound. As before, the subscripts ‘FX’ and ‘FR’ denote selection of the elements associated with the fixed and free variables.
Let CC denote the sub-matrix of rows of
(AL)
AN
AL AN
corresponding to general constraints in the working set. The general constraints in the working set will remain unaltered if
CFRdFR = 0,
CFRdFR=0,
(9)
which is equivalent to defining dFRdFR as
dFR = ZdZ
dFR=ZdZ
(10)
for some vector dZdZ, where ZZ is the matrix associated with the TQTQ factorization (5) of CFRCFR.
The definition of dZdZ in (10) depends on whether the current pp is feasible. If not, dZdZ is zero except for an element γγ in the jjth position, where jj and γγ are chosen so that the sum of infeasibilities is decreasing along dd. (For further details, see Gill et al. (1986).) In the feasible case, dZdZ satisfies the equations
RZT RZ dZ = ZT qFR ,
RZT RZ dZ = - ZT qFR ,
(11)
where RZRZ is the Cholesky factor of ZTHFRZZTHFRZ and qq is the gradient of the quadratic objective function (q = g + Hp)(q=g+Hp). (The vector ZTqFRZTqFR is the projected gradient of the QP.) With (11), p + dp+d is the minimizer of the quadratic objective function subject to treating the constraints in the working set as equalities.
If the QP projected gradient is zero, the current point is a constrained stationary point in the subspace defined by the working set. During the feasibility phase, the projected gradient will usually be zero only at a vertex (although it may vanish at non-vertices in the presence of constraint dependencies). During the optimality phase, a zero projected gradient implies that pp minimizes the quadratic objective function when the constraints in the working set are treated as equalities. In either case, Lagrange multipliers are computed. Given a positive constant δδ of the order of the machine precision, the Lagrange multiplier μjμj corresponding to an inequality constraint in the working set is said to be optimal if μjδμjδ when the jjth constraint is at its upper bound, or if μjδμj-δ when the associated constraint is at its lower bound. If any multiplier is nonoptimal, the current objective function (either the true objective or the sum of infeasibilities) can be reduced by deleting the corresponding constraint from the working set.
If optimal multipliers occur during the feasibility phase and the sum of infeasibilities is nonzero, no feasible point exists. The QP algorithm will then continue iterating to determine the minimum sum of infeasibilities. At this point, the Lagrange multiplier μjμj will satisfy (1 + δ)μjδ-(1+δ)μjδ for an inequality constraint at its upper bound, and δμj(1 + δ)-δμj(1+δ) for an inequality at its lower bound. The Lagrange multiplier for an equality constraint will satisfy |μj|1 + δ|μj|1+δ.
The choice of step length σσ in the QP iteration (8) is based on remaining feasible with respect to the satisfied constraints. During the optimality phase, if p + dp+d is feasible, σσ will be taken as unity. (In this case, the projected gradient at pp- will be zero.) Otherwise, σσ is set to σMσM, the step to the ‘nearest’ constraint, which is added to the working set at the next iteration.
Each change in the working set leads to a simple change to CFRCFR: if the status of a general constraint changes, a row of CFRCFR is altered; if a bound constraint enters or leaves the working set, a column of CFRCFR changes. Explicit representations are recurred of the matrices TT, QFRQFR and RR, and of the vectors QTqQTq and QTgQTg.

The Merit Function

After computing the search direction as described in Section [Solution of the Quadratic Programming Subproblem], each major iteration proceeds by determining a step length αα in (3) that produces a ‘sufficient decrease’ in the augmented Lagrangian merit function
L(x,λ,s) = F(x) λi(ci(x)si) + (1/2) ρi (ci(x)si) 2,
i i
L (x,λ,s) = F(x) - i λi ( ci (x) - si ) + 12 i ρi ( ci(x) - si ) 2 ,
(12)
where xx, λλ and ss vary during the linesearch. The summation terms in (12) involve only the nonlinear constraints. The vector λλ is an estimate of the Lagrange multipliers for the nonlinear constraints of (1). The non-negative slack variables {si}{si} allow nonlinear inequality constraints to be treated without introducing discontinuities. The solution of the QP subproblem (4) provides a vector triple that serves as a direction of search for the three sets of variables. The non-negative vector ρρ of penalty parameters is initialized to zero at the beginning of the first major iteration. Thereafter, selected elements are increased whenever necessary to ensure descent for the merit function. Thus, the sequence of norms of ρρ (the printed quantity Penalty; see Section [Description of Monitoring Information]) is generally nondecreasing, although each ρiρi may be reduced a limited number of times.
The merit function (12) and its global convergence properties are described in Gill et al. (1986a).

The Quasi-Newton Update

The matrix HH in (4) is a positive definite quasi-Newton approximation to the Hessian of the Lagrangian function. (For a review of quasi-Newton methods, see Dennis and Schnabel (1983).) At the end of each major iteration, a new Hessian approximation HH- is defined as a rank-two modification of HH. In nag_opt_nlp1_rcomm (e04uf), the BFGS (Broyden–Fletcher–Goldfarb–Shanno) quasi-Newton update is used:
H = H1/(sTHs)HssTH + 1/(yTs)yyT,
H-=H-1sTHs HssTH+1yTs yyT,
(13)
where s = xxs=x--x (the change in xx).
In nag_opt_nlp1_rcomm (e04uf), HH is required to be positive definite. If HH is positive definite, HH- defined by (13) will be positive definite if and only if yTsyTs is positive (see Dennis and Moré (1977)). Ideally, yy in (13) would be taken as yLyL, the change in gradient of the Lagrangian function
yL = g ANT μN g + ANT μN ,
yL = g- - A-NT μN - g + ANT μN ,
(14)
where μNμN denotes the QP multipliers associated with the nonlinear constraints of the original problem. If yLT s yLT s  is not sufficiently positive, an attempt is made to perform the update with a vector yy of the form
mN
y = yL + ωi(ai()ci()ai(x)ci(x)),
i = 1
y=yL+i=1mNωi(ai(x^)ci(x^)-ai(x)ci(x)),
where ωi0ωi0. If no such vector can be found, the update is performed with a scaled yLyL. In this case, M is printed to indicate that the update was modified.
Rather than modifying HH itself, the Cholesky factor of the transformed Hessian HQHQ (6) is updated, where QQ is the matrix from (5) associated with the active set of the QP subproblem. The update (13) is equivalent to the following update to HQHQ:
HQ = HQ 1/( sQT HQ sQ ) HQ sQ sQT HQ + 1/( yQT sQ ) yQ yQT ,
H-Q = HQ - 1 sQT HQ sQ HQ sQ sQT HQ + 1 yQT sQ yQ yQT ,
(15)
where yQ = QTyyQ=QTy, and sQ = QTssQ=QTs. This update may be expressed as a rank-one update to RR (see Dennis and Schnabel (1981)).

Optional Parameters

Several optional parameters in nag_opt_nlp1_rcomm (e04uf) define choices in the problem specification or the algorithm logic. In order to reduce the complexity of using nag_opt_nlp1_rcomm (e04uf) these optional parameters have associated default values that are appropriate for most problems. Therefore you need only specify those optional parameters whose values are to be different from their default values.
The remainder of this section can be skipped if you wish to use the default values for all optional parameters.
The following is a list of the optional parameters available. A full description of each optional parameter is provided in Section [Description of the optional parameters].
Optional parameters may be specified by calling nag_opt_nlp1_option_string (e04ue) before a call to nag_opt_nlp1_rcomm (e04uf).
nag_opt_nlp1_option_string (e04ue) can be called to supply options directly, one call being necessary for each optional parameter. For example,
[lwsav, iwsav, rwsav, inform] = e04ue('Print Level = 1', lwsav, iwsav, rwsav);
nag_opt_nlp1_option_string (e04ue) should be consulted for a full description of this method of supplying optional parameters.
All optional parameters not specified are set to their default values. Optional parameters specified are unaltered by nag_opt_nlp1_rcomm (e04uf) (unless they define invalid values) and so remain in effect for subsequent calls to nag_opt_nlp1_rcomm (e04uf).

Description of the Optional Parameters

For each option, we give a summary line, a description of the optional parameter and details of constraints.
The summary line contains:
Keywords and character values are case and white space insensitive.
Central Difference Interval  rr
Default values are computed
If the algorithm switches to central differences because the forward-difference approximation is not sufficiently accurate then the value of rr is used as the difference interval for every element of xx. The switch to central differences is indicated by C at the end of each line of intermediate printout produced by the major iterations (see Section [Printed output]). The use of finite differences is discussed further under the optional parameter Difference Interval.
If you supply a value for this optional parameter, a small value between 0.00.0 and 1.01.0 is appropriate.
Cold Start  
Default
Warm Start  
This option controls the specification of the initial working set in both the procedure for finding a feasible point for the linear constraints and bounds and in the first QP subproblem thereafter. With a Cold Start, the first working set is chosen by nag_opt_nlp1_rcomm (e04uf) based on the values of the variables and constraints at the initial point. Broadly speaking, the initial working set will include equality constraints and bounds or inequality constraints that violate or ‘nearly’ satisfy their bounds (to within Crash Tolerance).
With a Warm Start, you must set the istate array and define clamda and r as discussed in Section [Parameters]. istate values associated with bounds and linear constraints determine the initial working set of the procedure to find a feasible point with respect to the bounds and linear constraints. istate values associated with nonlinear constraints determine the initial working set of the first QP subproblem after such a feasible point has been found. nag_opt_nlp1_rcomm (e04uf) will override your specification of istate if necessary, so that a poor choice of the working set will not cause a fatal error. For instance, any elements of istate which are set to 2-2, 1​ or ​4-1​ or ​4 will be reset to zero, as will any elements which are set to 33 when the corresponding elements of bl and bu are not equal. A warm start will be advantageous if a good estimate of the initial working set is available – for example, when nag_opt_nlp1_rcomm (e04uf) is called repeatedly to solve related problems.
Crash Tolerance  rr
Default = 0.01=0.01
This value is used in conjunction with the optional parameter Cold Start (the default value) when nag_opt_nlp1_rcomm (e04uf) selects an initial working set. If 0r10r1, the initial working set will include (if possible) bounds or general inequality constraints that lie within rr of their bounds. In particular, a constraint of the form ajT xl ajT xl  will be included in the initial working set if |ajTxl| r (1 + |l|) | ajT x-l | r (1+|l|) . If r < 0r<0 or r > 1r>1, the default value is used.
Defaults  
This special keyword may be used to reset all optional parameters to their default values.
Derivative Level  ii
Default = 3=3
This parameter indicates which derivatives are provided during intermediate exits. The possible choices for ii are the following.
ii Meaning
3 All elements of the objective gradient and the constraint Jacobian are provided.
2 All elements of the constraint Jacobian are provided, but some elements of the objective gradient are not specified.
1 All elements of the objective gradient are provided, but some elements of the constraint Jacobian are not specified.
0 Some elements of both the objective gradient and the constraint Jacobian are not specified.
The value i = 3i=3 should be used whenever possible, since nag_opt_nlp1_rcomm (e04uf) is more reliable (and will usually be more efficient) when all derivatives are exact.
If i = 0​ or ​2i=0​ or ​2, nag_opt_nlp1_rcomm (e04uf) will estimate the unspecified elements of the objective gradient, using finite differences. The computation of finite difference approximations usually increases the total run-time, since an intermediate exit to the calling program is required for each unspecified element. Furthermore, less accuracy can be attained in the solution (see Chapter 8 of Gill et al. (1981), for a discussion of limiting accuracy).
If i = 0​ or ​1i=0​ or ​1, nag_opt_nlp1_rcomm (e04uf) will approximate unspecified elements of the constraint Jacobian. One intermediate exit is needed for each variable for which partial derivatives are not available. For example, if the Jacobian has the form
  * * * * * ? ? * * * ? * * * * *  
* * * * * ? ? * * * ? * * * * *
where ‘**’ indicates an element provided and ‘?’ indicates an unspecified element, nag_opt_nlp1_rcomm (e04uf) will make an intermediate exit to the calling program twice: once to estimate the missing element in column 22, and again to estimate the two missing elements in column 33. (Since columns 11 and 44 are known, they require no intermediate exits for information.)
At times, central differences are used rather than forward differences, in which case twice as many intermediate exits are needed. (The switch to central differences is not under your control.)
If i < 0i<0 or i > 3i>3, the default value is used.
Difference Interval  rr
Default values are computed
This option defines an interval used to estimate derivatives by finite differences in the following circumstances:
(a) For verifying the objective and/or constraint gradients (see the description of the optional parameter Verify).
(b) For estimating unspecified elements of the objective gradient or the constraint Jacobian.
In general, a derivative with respect to the jjth variable is approximated using the interval δjδj, where δj = r(1 + |j|)δj=r(1+|x^j|), with x^ the first point feasible with respect to the bounds and linear constraints. If the functions are well scaled then the resulting derivative approximation should be accurate to O(r)O(r). See Gill et al. (1981) for a discussion of the accuracy in finite difference approximations.
If a difference interval is not specified then a finite difference interval will be computed automatically for each variable by a procedure that requires up to six intermediate exits for each element. This option is recommended if the function is badly scaled or you wish to have nag_opt_nlp1_rcomm (e04uf) determine constant elements in the objective and constraint gradients.
If you supply a value for this optional parameter, a small value between 0.00.0 and 1.01.0 is appropriate.
Feasibility Tolerance  rr
Default = sqrt(ε)=ε
The scalar rr defines the maximum acceptable absolute violations in linear and nonlinear constraints at a ‘feasible’ point; i.e., a constraint is considered satisfied if its violation does not exceed rr. If r < εr<ε or r1r1, the default value is used. Using this keyword sets both optional parameters Linear Feasibility Tolerance and Nonlinear Feasibility Tolerance to rr, if εr < 1εr<1. (Additional details are given under the descriptions of these optional parameters.)
Function Precision  rr
Default = ε0.9=ε0.9
This parameter defines εrεr, which is intended to be a measure of the accuracy with which the problem functions F(x)F(x) and c(x)c(x) can be computed. If r < εr<ε or r1r1, the default value is used.
The value of εrεr should reflect the relative precision of 1 + |F(x)|1+|F(x)|; i.e., εrεr acts as a relative precision when |F||F| is large and as an absolute precision when |F||F| is small. For example, if F(x)F(x) is typically of order 10001000 and the first six significant digits are known to be correct, an appropriate value for εrεr would be 10610-6. In contrast, if F(x)F(x) is typically of order 10410-4 and the first six significant digits are known to be correct, an appropriate value for εrεr would be 101010-10. The choice of εrεr can be quite complicated for badly scaled problems; see Chapter 8 of Gill et al. (1981) for a discussion of scaling techniques. The default value is appropriate for most simple functions that are computed with full accuracy. However, when the accuracy of the computed function values is known to be significantly worse than full precision, the value of εrεr should be large enough so that nag_opt_nlp1_rcomm (e04uf) will not attempt to distinguish between function values that differ by less than the error inherent in the calculation.
Hessian  
Default = NO=NO
This option controls the contents of the upper triangular matrix RR (see Section [Parameters]). nag_opt_nlp1_rcomm (e04uf) works exclusively with the transformed and reordered Hessian HQHQ (6), and hence extra computation is required to form the Hessian itself. If Hessian = NOHessian=NO, r contains the Cholesky factor of the transformed and reordered Hessian. If Hessian = YESHessian=YES, the Cholesky factor of the approximate Hessian itself is formed and stored in r. You should select Hessian = YESHessian=YES if a Warm Start will be used for the next call to nag_opt_nlp1_rcomm (e04uf).
Infinite Bound Size  rr
Default = 1020=1020
If r > 0r>0, rr defines the ‘infinite’ bound bigbndbigbnd in the definition of the problem constraints. Any upper bound greater than or equal to bigbndbigbnd will be regarded as + + (and similarly any lower bound less than or equal to bigbnd-bigbnd will be regarded as -). If r < 0r<0, the default value is used.
Infinite Step Size  rr
Default = max (bigbnd,1020)=max(bigbnd,1020)
If r > 0r>0, rr specifies the magnitude of the change in variables that is treated as a step to an unbounded solution. If the change in xx during an iteration would exceed the value of rr, the objective function is considered to be unbounded below in the feasible region. If r0r0, the default value is used.
Line Search Tolerance  rr
Default = 0.9=0.9
The value rr (0r < 10r<1) controls the accuracy with which the step αα taken during each iteration approximates a minimum of the merit function along the search direction (the smaller the value of rr, the more accurate the linesearch). The default value r = 0.9r=0.9 requests an inaccurate search and is appropriate for most problems, particularly those with any nonlinear constraints.
If there are no nonlinear constraints, a more accurate search may be appropriate when it is desirable to reduce the number of major iterations – for example, if the objective function is cheap to evaluate, or if a substantial number of derivatives are unspecified. If r < 0r<0 or r1r1, the default value is used.
Linear Feasibility Tolerance  r1r1
Default = sqrt(ε)=ε
Nonlinear Feasibility Tolerance  r2r2
Default = ε0.33=ε0.33 or sqrt(ε)ε
The default value of r2r2 is ε0.33ε0.33 if Derivative Level = 0Derivative Level=0 or 11, and sqrt(ε)ε otherwise.
The scalars r1r1 and r2r2 define the maximum acceptable absolute violations in linear and nonlinear constraints at a ‘feasible’ point; i.e., a linear constraint is considered satisfied if its violation does not exceed r1r1. Similarly a nonlinear constraint is considered satisfied if its violation does not exceed r2r2. If rm < εrm<ε or rm1rm1, the default value is used, for m = 1,2m=1,2.
On entry to nag_opt_nlp1_rcomm (e04uf), an iterative procedure is executed in order to find a point that satisfies the linear constraints and bounds on the variables to within the tolerance r1r1. All subsequent iterates will satisfy the linear constraints to within the same tolerance (unless r1r1 is comparable to the finite difference interval).
For nonlinear constraints, the feasibility tolerance r2r2 defines the largest constraint violation that is acceptable at an optimal point. Since nonlinear constraints are generally not satisfied until the final iterate, the value of optional parameter Nonlinear Feasibility Tolerance acts as a partial termination criterion for the iterative sequence generated by nag_opt_nlp1_rcomm (e04uf) (see the discussion of optional parameter Optimality Tolerance).
These tolerances should reflect the precision of the corresponding constraints. For example, if the variables and the coefficients in the linear constraints are of order unity, and the latter are correct to about 66 decimal digits, it would be appropriate to specify r1r1 as 10610-6.
List  
Nolist  
Default for nag_opt_nlp1_rcomm (e04uf)
Optional parameter List may be used to turn on printing of each optional parameter specification as it is supplied. Nolist may then be used to suppress this printing again.
Major Iteration Limit  ii
Default = max (50,3(n + nL) + 10nN)=max(50,3(n+nL)+10nN)
Iteration Limit  
Iters  
Itns  
The value of ii specifies the maximum number of major iterations allowed before termination. Setting i = 0i=0 and Major Print Level > 0Major Print Level>0 means that the workspace needed will be computed and printed, but no iterations will be performed. If i < 0i<0, the default value is used.
Major Print Level  ii
Print Level  
Default for nag_opt_nlp1_rcomm (e04uf) = 0=0
The value of ii controls the amount of printout produced by the major iterations of nag_opt_nlp1_rcomm (e04uf), as indicated below. A detailed description of the printed output is given in Section [Printed output] (summary output at each major iteration and the final solution) and Section [Description of Monitoring Information] (monitoring information at each major iteration). (See also the description of the optional parameter Minor Print Level.)
The following printout is sent to the current advisory message unit (as defined by nag_file_set_unit_advisory (x04ab)):
ii Output
0000 No output.
0101 The final solution only.
0505 One line of summary output ( < 80<80 characters; see Section [Printed output]) for each major iteration (no printout of the final solution).
1010 The final solution and one line of summary output for each major iteration.
The following printout is sent to the logical unit number by the optional parameter Monitoring File:
ii Output
< 5<5 No output.
55 One long line of output ( > 80>80 characters; see Section [Description of Monitoring Information]) for each major iteration (no printout of the final solution).
2020 At each major iteration, the objective function, the Euclidean norm of the nonlinear constraint violations, the values of the nonlinear constraints (the vector cc), the values of the linear constraints (the vector ALxALx) and the current values of the variables (the vector xx).
3030 At each major iteration, the diagonal elements of the matrix TT associated with the TQTQ factorization (5) (see Section [Overview]) of the QP working set and the diagonal elements of RR, the triangular factor of the transformed and reordered Hessian (6) (see Section [Overview]).
If Major Print Level5Major Print Level5 and the unit number defined by the optional parameter Monitoring File is the same as that defined by nag_file_set_unit_advisory (x04ab), then the summary output for each major iteration is suppressed.
Minor Iteration Limit  ii
Default = max (50,3(n + nL + nN))=max(50,3(n+nL+nN))
The value of ii specifies the maximum number of iterations for finding a feasible point with respect to the bounds and linear constraints (if any). The value of ii also specifies the maximum number of minor iterations for the optimality phase of each QP subproblem. If i0i0, the default value is used.
Minor Print Level  ii
Default = 0=0
The value of ii controls the amount of printout produced by the minor iterations of nag_opt_nlp1_rcomm (e04uf) (i.e., the iterations of the quadratic programming algorithm), as indicated below. A detailed description of the printed output is given in Section [Printed output] in (e04nc) (summary output at each minor iteration and the final QP solution) and Section [Description of Monitoring Information] in (e04nc)) (monitoring information at each minor iteration). (See also the description of the optional parameter Major Print Level.)
The following printout is sent to the current advisory message unit (as defined by nag_file_set_unit_advisory (x04ab)):
ii Output
0000 No output.
0101 The final QP solution only.
0505 One line of summary output ( < 80<80 characters; see Section [Printed output] in (e04nc)) for each minor iteration (no printout of the final QP solution).
1010 The final QP solution and one line of summary output for each minor iteration.
The following printout is sent to the logical unit number defined by the optional parameter Monitoring File:
ii Output
< 5<5 No output.
55 One long line of output ( > 80>80 characters; see Section [Printed output] in (e04nc)) for each minor iteration (no printout of the final QP solution).
2020 At each minor iteration, the current estimates of the QP multipliers, the current estimate of the QP search direction, the QP constraint values and the status of each QP constraint.
3030 At each minor iteration, the diagonal elements of the matrix TT associated with the TQTQ factorization (5) (see Section [Overview]) of the QP working set and the diagonal elements of the Cholesky factor RR of the transformed Hessian (6) (see Section [Overview]).
If Minor Print Level5Minor Print Level5 and the unit number defined by the optional parameter Monitoring File is the same as that defined by nag_file_set_unit_advisory (x04ab) then the summary output for each minor iteration is suppressed.
Monitoring File  ii
Default = 1=-1
If i0i0 and Major Print Level5Major Print Level5 or i0i0 and Minor Print Level5Minor Print Level5, monitoring information produced by nag_opt_nlp1_rcomm (e04uf) at every iteration is sent to a file with logical unit number ii. If i < 0i<0 and/or Major Print Level < 5Major Print Level<5 and Minor Print Level < 5Minor Print Level<5, no monitoring information is produced.
Optimality Tolerance  rr
Default = εr0.8=εr0.8
The parameter rr (εrr < 1εrr<1) specifies the accuracy to which you wish the final iterate to approximate a solution of the problem. Broadly speaking, rr indicates the number of correct figures desired in the objective function at the solution. For example, if rr is 10610-6 and nag_opt_nlp1_rcomm (e04uf) terminates successfully, the final value of FF should have approximately six correct figures. If r < εrr<εr or r1r1, the default value is used.
nag_opt_nlp1_rcomm (e04uf) will terminate successfully if the iterative sequence of xx values is judged to have converged and the final point satisfies the first-order Kuhn–Tucker conditions (see Section [Overview]). The sequence of iterates is considered to have converged at xx if
αpsqrt(r)(1 + x),
αpr(1+x),
(16)
where pp is the search direction and αα the step length from (3). An iterate is considered to satisfy the first-order conditions for a minimum if
ZTgFR sqrt( r ) (1 + max ( 1 + |F(x)| ,gFR))
ZT g FR r ( 1 + max( 1 + | F ( x ) | , g FR ) )
(17)
and
|resj|ftol  for all  j,
|resj|ftol  for all  j,
(18)
where ZTgFRZTgFR is the projected gradient (see Section [Overview]), gFRgFR is the gradient of F(x)F(x) with respect to the free variables, resjresj is the violation of the jjth active nonlinear constraint, and ftolftol is the Nonlinear Feasibility Tolerance.
Start Objective Check At Variable  i1i1
Default = 1=1
Stop Objective Check At Variable  i2i2
Default = n=n
Start Constraint Check At Variable  i3i3
Default = 1=1
Stop Constraint Check At Variable  i4i4
Default = n=n
These keywords take effect only if Verify Level > 0Verify Level>0. They may be used to control the verification of gradient elements and/or Jacobian elements computed by the calling program during intermediate exits. For example, if the first 3030 elements of the objective gradient appeared to be correct in an earlier run, so that only element 3131 remains questionable, it is reasonable to specify Start Objective Check At Variable = 31Start Objective Check At Variable=31. If the first 3030 variables appear linearly in the objective, so that the corresponding gradient elements are constant, the above choice would also be appropriate.
If i2m10i2m-10 or i2m1 > min (n,i2m)i2m-1>min(n,i2m), the default value is used, for m = 1,2m=1,2. If i2m0i2m0 or i2m > ni2m>n, the default value is used, for m = 1,2m=1,2.
Step Limit  rr
Default = 2.0=2.0
If r > 0,rr>0,r specifies the maximum change in variables at the first step of the linesearch. In some cases, such as F(x) = aebxF(x)=aebx or F(x) = axbF(x)=axb, even a moderate change in the elements of xx can lead to floating point overflow. The parameter rr is therefore used to encourage evaluation of the problem functions at meaningful points. Given any major iterate xx, the first point x~ at which FF and cc are evaluated during the linesearch is restricted so that
x2r(1 + x2).
x~-x2r(1+x2).
The linesearch may go on and evaluate FF and cc at points further from xx if this will result in a lower value of the merit function (indicated by L at the end of each line of output produced by the major iterations; see Section [Printed output]). If L is printed for most of the iterations, rr should be set to a larger value.
Wherever possible, upper and lower bounds on xx should be used to prevent evaluation of nonlinear functions at wild values. The default value Step Limit = 2.0Step Limit=2.0 should not affect progress on well-behaved functions, but values such as 0.1 ​ or ​ 0.010.1 ​ or ​ 0.01 may be helpful when rapidly varying functions are present. If a small value of Step Limit is selected then a good starting point may be required. An important application is to the class of nonlinear least squares problems. If r0r0, the default value is used.
Verify Level  ii
Default = 0=0
Verify  
Verify Constraint Gradients  
Verify Gradients  
Verify Objective Gradients  
These keywords refer to finite difference checks on the gradient elements computed by the calling program during intermediate exits. (Unspecified gradient elements are not checked.) The possible choices for ii are as follows:
ii Meaning
1-1 No checks are performed.
0-0 Only a ‘cheap’ test will be performed.
11 In addition to the ‘cheap’ test, individual gradient elements will also be checked using a reliable (but more expensive) test.
It is possible to specify Verify Level = 0Verify Level=0 to 33 in several ways. For example, the objective gradient will be verified if Verify, Verify = YESVerify=YES, Verify Gradients, Verify Objective Gradients or Verify Level = 1Verify Level=1 is specified. The constraint gradients will be verified if Verify = YESVerify=YES or Verify Level = 2Verify Level=2 or Verify is specified. Similarly, the objective and the constraint gradients will be verified if Verify = YESVerify=YES or Verify Level = 3Verify Level=3 or Verify is specified.
If 0i30i3, gradients will be verified at the first point that satisfies the linear constraints and bounds.
If i = 0i=0, only a ‘cheap’ test will be performed, requiring one intermediate exit for the objective function gradients and (if appropriate) one intermediate exit for the partial derivatives of the constraints.
If 1i31i3, a more reliable (but more expensive) check will be made on individual gradient elements, within the ranges specified by the Start Objective Check At Variable and Stop Objective Check At Variable keywords. A result of the form OK or BAD? is printed by nag_opt_nlp1_rcomm (e04uf) to indicate whether or not each element appears to be correct.
If 10i1310i13, the action is the same as for i < 10i<10, except that it will take place at the user-specified initial value of xx.
If i < 1i<-1 or 4i94i9 or i > 13i>13, the default value is used.
We suggest that Verify Level = 3Verify Level=3 be used whenever a new calling program is being developed.

Description of Monitoring Information

This section describes the long line of output ( > 80>80 characters) which forms part of the monitoring information produced by nag_opt_nlp1_rcomm (e04uf). (See also the description of the optional parameters Major Print Level, Minor Print Level and Monitoring File.) You can control the level of printed output (see the description of the optional parameter Major Print Level).
When Major Print Level5Major Print Level5 and Monitoring File0Monitoring File0, the following line of output is produced at every major iteration of nag_opt_nlp1_rcomm (e04uf) on the unit number specified by Monitoring File. In all cases, the values of the quantities printed are those in effect on completion of the given iteration.
Maj is the major iteration count.
Mnr is the number of minor iterations required by the feasibility and optimality phases of the QP subproblem. Generally, Mnr will be 11 in the later iterations, since theoretical analysis predicts that the correct active set will be identified near the solution (see Section [Algorithmic Details]).
Note that Mnr may be greater than the optional parameter Minor Iteration Limit if some iterations are required for the feasibility phase.
Step is the step αkαk taken along the computed search direction. On reasonably well-behaved problems, the unit step (i.e., αk = 1αk=1) will be taken as the solution is approached.
Nfun is the cumulative number of evaluations of the objective function needed for the linesearch. Evaluations needed for the estimation of the gradients by finite differences are not included. Nfun is printed as a guide to the amount of work required for the linesearch.
Merit Function is the value of the augmented Lagrangian merit function (12) at the current iterate. This function will decrease at each iteration unless it was necessary to increase the penalty parameters (see Section [The Merit Function]). As the solution is approached, Merit Function will converge to the value of the objective function at the solution.
If the QP subproblem does not have a feasible point (signified by I at the end of the current output line) then the merit function is a large multiple of the constraint violations, weighted by the penalty parameters. During a sequence of major iterations with infeasible subproblems, the sequence of Merit Function values will decrease monotonically until either a feasible subproblem is obtained or nag_opt_nlp1_rcomm (e04uf) terminates with ifail = 3ifail=3 (no feasible point could be found for the nonlinear constraints).
If there are no nonlinear constraints present (i.e., ncnln = 0ncnln=0) then this entry contains Objective, the value of the objective function F(x)F(x). The objective function will decrease monotonically to its optimal value when there are no nonlinear constraints.
Norm Gz is ZTgFRZTgFR, the Euclidean norm of the projected gradient (see Section [Solution of the Quadratic Programming Subproblem]). Norm Gz will be approximately zero in the neighbourhood of a solution.
Violtn is the Euclidean norm of the residuals of constraints that are violated or in the predicted active set (not printed if ncnln is zero). Violtn will be approximately zero in the neighbourhood of a solution.
Nz is the number of columns of ZZ (see Section [Solution of the Quadratic Programming Subproblem]). The value of Nz is the number of variables minus the number of constraints in the predicted active set; i.e., Nz = n(Bnd + Lin + Nln)Nz=n-(Bnd+Lin+Nln).
Bnd is the number of simple bound constraints in the predicted active set.
Lin is the number of general linear constraints in the predicted working set.
Nln is the number of nonlinear constraints in the predicted active set (not printed if ncnln is zero).
Penalty is the Euclidean norm of the vector of penalty parameters used in the augmented Lagrangian merit function (not printed if ncnln is zero).
Cond H is a lower bound on the condition number of the Hessian approximation HH.
Cond Hz is a lower bound on the condition number of the projected Hessian approximation HZ HZ  ( HZ = ZT HFR Z = RZT RZ HZ = ZT HFR Z = RZT RZ ; see (6)). The larger this number, the more difficult the problem.
Cond T is a lower bound on the condition number of the matrix of predicted active constraints.
Conv is a three-letter indication of the status of the three convergence tests (16)(18) defined in the description of the optional parameter Optimality Tolerance. Each letter is T if the test is satisfied and F otherwise. The three tests indicate whether:
(i) the sequence of iterates has converged;
(ii) the projected gradient (Norm Gz) is sufficiently small; and
(iii) the norm of the residuals of constraints in the predicted active set (Violtn) is small enough.
If any of these indicators is F when nag_opt_nlp1_rcomm (e04uf) terminates with ifail = 0ifail=0, you should check the solution carefully.
M is printed if the quasi-Newton update has been modified to ensure that the Hessian approximation is positive definite (see Section [The Quasi-Newton Update]).
I is printed if the QP subproblem has no feasible point.
C is printed if central differences have been used to compute the unspecified objective and constraint gradients. If the value of Step is zero then the switch to central differences was made because no lower point could be found in the linesearch. (In this case, the QP subproblem is resolved with the central difference gradient and Jacobian.) If the value of Step is nonzero then central differences were computed because Norm Gz and Violtn imply that xx is close to a Kuhn–Tucker point (see Section [Overview]).
L is printed if the linesearch has produced a relative change in xx greater than the value defined by the optional parameter Step Limit. If this output occurs frequently during later iterations of the run, optional parameter Step Limit should be set to a larger value.
Need not be initialized if the (default) optional parameter Cold Start is used.
If the optional parameter Warm Start has been chosen, r must contain the upper triangular Cholesky factor RR of the initial approximation of the Hessian of the Lagrangian function, with the variables in the natural order. Elements not in the upper triangular part of r are assumed to be zero and need not be assigned.

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Chapter Contents
Chapter Introduction
NAG Toolbox

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