hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_opt_lsq_lincon_solve (e04nc)

Purpose

nag_opt_lsq_lincon_solve (e04nc) solves linearly constrained linear least squares problems and convex quadratic programming problems. It is not intended for large sparse problems.

Syntax

[istate, kx, x, a, b, iter, obj, clamda, lwsav, iwsav, rwsav, ifail] = e04nc(c, bl, bu, cvec, istate, kx, x, a, b, lwsav, iwsav, rwsav, 'm', m, 'n', n, 'nclin', nclin)
[istate, kx, x, a, b, iter, obj, clamda, lwsav, iwsav, rwsav, ifail] = nag_opt_lsq_lincon_solve(c, bl, bu, cvec, istate, kx, x, a, b, lwsav, iwsav, rwsav, 'm', m, 'n', n, 'nclin', nclin)
Before calling nag_opt_lsq_lincon_solve (e04nc), or the option setting function (e04ne), nag_opt_init (e04wb) must be called.

Description

nag_opt_lsq_lincon_solve (e04nc) is designed to solve a class of quadratic programming problems of the following general form:
minimize F(x)  subject to  l{
x
Cx
}
u
xRn
minimize xRn F(x)  subject to  l{ x Cx } u
(1)
where cc is an nLnL by nn matrix and the objective function F(x)F(x) may be specified in a variety of ways depending upon the particular problem to be solved. The available forms for F(x)F(x) are listed in Table 1, in which the prefixes FP, LP, QP and LS stand for ‘feasible point’, ‘linear programming’, ‘quadratic programming’ and ‘least squares’ respectively, cc is an nn-element vector, bb is an mm element vector and zz denotes the Euclidean length of zz.
Problem type F(x)F(x) Matrix AA
FP None Not applicable
LP cTxcTx Not applicable
QP1 cTx + (1/2)xTAxcTx+12xTAx nn by nn symmetric positive semidefinite
QP2 cTx + (1/2)xTAxcTx+12xTAx nn by nn symmetric positive semidefinite
QP3 cTx + (1/2)xTATAxcTx+12xTATAx mm by nn upper trapezoidal
QP4 cTx + (1/2)xTATAxcTx+12xTATAx mm by nn upper trapezoidal
LS1 cTx + (1/2)bAx2cTx+12b-Ax2 mm by nn
LS2 cTx + (1/2)bAx2cTx+12b-Ax2 mm by nn
LS3 cTx + (1/2)bAx2cTx+12b-Ax2 mm by nn upper trapezoidal
LS4 cTx + (1/2)bAx2cTx+12b-Ax2 mm by nn upper trapezoidal
Table 1
In the standard LS problem F(x)F(x) will usually have the form LS1, and in the standard convex QP problem F(x)F(x) will usually have the form QP2. The default problem type is LS1 and other objective functions are selected by using the optional parameter Problem Type.
When AA is upper trapezoidal it will usually be the case that m = nm=n, so that AA is upper triangular, but full generality has been allowed for in the specification of the problem. The upper trapezoidal form is intended for cases where a previous factorization, such as a QRQR factorization, has been performed.
The constraints involving cc are called the general constraints. Note that upper and lower bounds are specified for all the variables and for all the general 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.)
The defining feature of a quadratic function F(x)F(x) is that the second-derivative matrix HH (the Hessian matrix) is constant. For the LP case H = 0H=0; for QP1 and QP2, H = AH=A; for QP3 and QP4, H = ATAH=ATA and for LS1 (the default), LS2, LS3 and LS4, H = ATAH=ATA.
Problems of type QP3 and QP4 for which AA is not in upper trapezoidal form should be solved as types LS1 and LS2 respectively, with b = 0b=0.
For problems of type LS, we refer to AA as the least squares matrix, or the matrix of observations and to bb as the vector of observations.
You must supply an initial estimate of the solution.
If HH is nonsingular then nag_opt_lsq_lincon_solve (e04nc) will obtain the unique (global) minimum. If HH is singular then the solution may still be a global minimum if all active constraints have nonzero Lagrange multipliers. Otherwise the solution obtained will be either a weak minimum (i.e., with a unique optimal objective value, but an infinite set of optimal xx), or else the objective function is unbounded below in the feasible region. The last case can only occur when F(x)F(x) contains an explicit linear term (as in problems LP, QP2, QP4, LS2 and LS4).
The method used by nag_opt_lsq_lincon_solve (e04nc) is described in detail in Section [Algorithmic Details].

References

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 (1984) 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 and Wright M H (1981) Practical Optimization Academic Press
Stoer J (1971) On the numerical solution of constrained least squares problems SIAM J. Numer. Anal. 8 382–411

Parameters

Compulsory Input Parameters

1:     c(ldc, : :) – double array
The first dimension of the array c 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
The iith row of c must contain the coefficients of the iith general constraint, for i = 1,2,,nclini=1,2,,nclin.
If nclin = 0nclin=0, c is not referenced.
2:     bl(n + nclinn+nclin) – double array
3:     bu(n + nclinn+nclin) – double array
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, and the next nLnL elements must contain the bounds for the general linear 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 bu(j) = bl(j) = βbuj=blj=β, say, where |β| < bigbnd|β|<bigbnd.
Constraints:
  • bl(j)bu(j)bljbuj, for j = 1,2,,n + nclinj=1,2,,n+nclin;
  • if bl(j) = bu(j) = βblj=buj=β, |β| < bigbnd|β|<bigbnd.
4:     cvec( : :) – double array
Note: the dimension of the array cvec must be at least nn if the problem is of type LP, QP2, QP4, LS2 or LS4, and at least 11 otherwise.
The coefficients of the explicit linear term of the objective function.
If the problem is of type FP, QP1, QP3, LS1 (the default) or LS3, cvec is not referenced.
5:     istate(n + nclinn+nclin) – int64int32nag_int array
Need not be set if the (default) optional parameter Cold Start is used.
If the optional parameter Warm Start has been chosen, istate specifies the desired status of the constraints at the start of the feasibility phase. More precisely, the first nn elements of istate refer to the upper and lower bounds on the variables, and the next nLnL elements refer to the general linear constraints (if any). Possible values for istate(j)istatej are as follows:
istate(j)istatej Meaning
0 The constraint should not be in the initial working set.
1 The constraint should be in the initial working set at its lower bound.
2 The constraint should be in the initial working set at its upper bound.
3 The constraint should be in the initial working set as an equality. 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 reset to zero by the function. If nag_opt_lsq_lincon_solve (e04nc) has been called previously with the same values of n and nclin, 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 + nclinj=1,2,,n+nclin.
6:     kx(n) – int64int32nag_int array
n, the dimension of the array, must satisfy the constraint n > 0n>0.
Need not be initialized for problems of type FP, LP, QP1, QP2, LS1 (the default) or LS2.
For problems QP3, QP4, LS3 or LS4, kx must specify the order of the columns of the matrix AA with respect to the ordering of x. Thus if column jj of AA is the column associated with the variable xixi then kx(j) = ikxj=i.
Constraints:
  • 1kx(i)n1kxin, for i = 1,2,,ni=1,2,,n;
  • if ijij, kx(i)kx(j)kxikxj.
7:     x(n) – double array
n, the dimension of the array, must satisfy the constraint n > 0n>0.
An initial estimate of the solution.
Note: that it may be best to avoid the choice x = 0.0x=0.0.
8:     a(lda, : :) – double array
The first dimension of the array a must be at least max (1,m)max(1,m)
The second dimension of the array must be at least nn if the problem is of type QP1, QP2, QP3, QP4, LS1 (the default), LS2, LS3 or LS4, and at least 11 otherwise
The array a must contain the matrix AA as specified in Table 1 (see Section [Description]).
If the problem is of type QP1 or QP2, the first mm rows and columns of a must contain the leading mm by mm rows and columns of the symmetric Hessian matrix. Only the diagonal and upper triangular elements of the leading mm rows and columns of a are referenced. The remaining elements are assumed to be zero and need not be assigned.
For problems QP3, QP4, LS3 or LS4, the first mm rows of a must contain an mm by nn upper trapezoidal factor of either the Hessian matrix or the least squares matrix, ordered according to the kx array. The factor need not be of full rank, i.e., some of the diagonals may be zero. However, as a general rule, the larger the dimension of the leading nonsingular sub-matrix of AA, the fewer iterations will be required. Elements outside the upper triangular part of the first mm rows of a are assumed to be zero and need not be assigned.
If a constrained least squares problem contains a very large number of observations, storage limitations may prevent storage of the entire least squares matrix. In such cases, you should transform the original AA into a triangular matrix before the call to nag_opt_lsq_lincon_solve (e04nc) and solve the problem as type LS3 or LS4.
9:     b( : :) – double array
Note: the dimension of the array b must be at least mm if the problem is of type LS1 (the default), LS2, LS3 or LS4, and at least 11 otherwise.
The mm elements of the vector of observations.
10:   lwsav(120120) – logical array
11:   iwsav(610610) – int64int32nag_int array
12:   rwsav(475475) – double array
The arrays lwsav, iwsav and rwsav must not be altered between calls to any of the functions nag_opt_lsq_lincon_solve (e04nc), (e04nd) or (e04ne).

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The dimension of the array a.
mm, the number of rows in the matrix AA. If the problem is specified as type FP or LP, m is not referenced and is assumed to be zero.
If the problem is of type QP, m will usually be nn, the number of variables. However, a value of m less than nn is appropriate for QP3 or QP4 if AA is an upper trapezoidal matrix with mm rows. Similarly, m may be used to define the dimension of a leading block of nonzeros in the Hessian matrices of QP1 or QP2, in which case the last (nm)(n-m) rows and columns of a are assumed to be zero. In the QP case, mm should not be greater than nn; if it is, the last (mn)(m-n) rows of AA are ignored.
If the problem is of type LS1 (the default) or specified as type LS2, LS3 or LS4, m is also the dimension of the array b. Note that all possibilities (m < nm<n, m = nm=n and m > nm>n) are allowed in this case.
Constraint: m > 0m>0 if the problem is not of type FP or LP.
2:     n – int64int32nag_int scalar
Default: The dimension of the arrays kx, x. (An error is raised if these dimensions are not equal.)
nn, the number of variables.
Constraint: n > 0n>0.
3:     nclin – int64int32nag_int scalar
Default: The dimension of the array c.
nLnL, the number of general linear constraints.
Constraint: nclin0nclin0.

Input Parameters Omitted from the MATLAB Interface

ldc lda iwork liwork work lwork

Output Parameters

1:     istate(n + nclinn+nclin) – int64int32nag_int array
The status of the constraints in the 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 The constraint violates its lower bound by more than the feasibility tolerance.
1-1 The constraint violates its upper bound by more than the feasibility tolerance.
0-0 The constraint is satisfied to within the feasibility tolerance, but is not in the working set.
1-1 This inequality constraint is included in the working set at its lower bound.
2-2 This inequality constraint is included in the working set at its upper bound.
3-3 The constraint is included in the working set as an equality. This value of istate can occur only when bl(j) = bu(j)blj=buj.
4-4 This corresponds to optimality being declared with x(j)xj being temporarily fixed at its current value.
2:     kx(n) – int64int32nag_int array
Defines the order of the columns of a with respect to the ordering of x, as described above.
3:     x(n) – double array
The point at which nag_opt_lsq_lincon_solve (e04nc) terminated. If ifail = 0ifail=0, 11 or 44, x contains an estimate of the solution.
4:     a(lda, : :) – double array
The first dimension of the array a will be max (1,m)max(1,m)
The second dimension of the array will be nn if the problem is of type QP1, QP2, QP3, QP4, LS1 (the default), LS2, LS3 or LS4, and at least 11 otherwise
ldamax (1,m)ldamax(1,m).
If Hessian = NOHessian=NO and the problem is of type LS or QP, a contains the upper triangular Cholesky factor RR of (8) (see Section [Main Iteration]), with columns ordered as indicated by kx. If Hessian = YESHessian=YES and the problem is of type LS or QP, a contains the upper triangular Cholesky factor RR of the Hessian matrix HH, with columns ordered as indicated by kx. In either case RR may be used to obtain the variance-covariance matrix or to recover the upper triangular factor of the original least squares matrix.
If the problem is of type FP or LP, a is not referenced.
5:     b( : :) – double array
Note: the dimension of the array b must be at least mm if the problem is of type LS1 (the default), LS2, LS3 or LS4, and at least 11 otherwise.
The transformed residual vector of equation (10) (see Section [Main Iteration]).
If the problem is of type FP, LP, QP1, QP2, QP3 or QP4, b is not referenced.
6:     iter – int64int32nag_int scalar
The total number of iterations performed.
7:     obj – double scalar
The value of the objective function at xx if xx is feasible, or the sum of infeasibiliites at xx otherwise. If the problem is of type FP and xx is feasible, obj is set to zero.
8:     clamda(n + nclinn+nclin) – double array
The values of the Lagrange multipliers for each constraint with respect to the current working set. The first nn elements contain the multipliers for the bound constraints on the variables, and the next nLnL elements contain the multipliers for the general linear constraints (if any). If istate(j) = 0istatej=0 (i.e., constraint jj is not in the working set), clamda(j)clamdaj is zero. If xx is optimal, clamda(j)clamdaj should be non-negative if istate(j) = 1istatej=1, non-positive if istate(j) = 2istatej=2 and zero if istate(j) = 4istatej=4.
9:     lwsav(120120) – logical array
10:   iwsav(610610) – int64int32nag_int array
11:   rwsav(475475) – double array
12:   ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).
nag_opt_lsq_lincon_solve (e04nc) returns with ifail = 0ifail=0 if xx is a strong local minimizer, i.e., the projected gradient (Norm Gz; see Section [Printed output]) is negligible, the Lagrange multipliers (Lagr Mult; see Section [Definition of Search Direction]) are optimal and RZRZ (see Section [Main Iteration]) is nonsingular.

Error Indicators and Warnings

Note: nag_opt_lsq_lincon_solve (e04nc) 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 = 1ifail=1
x is a weak local minimum, (i.e., the projected gradient is negligible, the Lagrange multipliers are optimal, but either RZRZ (see Section [Main Iteration]) is singular, or there is a small multiplier). This means that xx is not unique.
W ifail = 2ifail=2
The solution appears to be unbounded. This value of ifail implies that a step as large as Infinite Bound Size (default value = 1020default value=1020) would have to be taken in order to continue the algorithm. This situation can occur only when AA is singular, there is an explicit linear term, and at least one variable has no upper or lower bound.
W ifail = 3ifail=3
No feasible point was found, i.e., it was not possible to satisfy all the constraints to within the feasibility tolerance. In this case, the constraint violations at the final xx will reveal a value of the tolerance for which a feasible point will exist – for example, when the feasibility tolerance for each violated constraint exceeds its Slack (see Section [Printed output]) at the final point. The modified problem (with an altered feasibility tolerance) may then be solved using a Warm Start. You should check that there are no constraint redundancies. If the data for the constraints are accurate only to the absolute precision σσ, you should ensure that the value of the optional parameter Feasibility Tolerance (default value = sqrt(ε)default value=ε, where εε is the machine precision) is greater than σσ. For example, if all elements of cc are of order unity and are accurate only to three decimal places, the Feasibility Tolerance should be at least 10310-3.
  ifail = 4ifail=4
The limiting number of iterations (determined by the optional parameters Feasibility Phase Iteration Limit (default value = max (50,5(n + nL))default value=max(50,5(n+nL))) and Optimality Phase Iteration Limit (default value = max (50,5(n + nL))default value=max(50,5(n+nL)))) was reached before normal termination occurred. If the method appears to be making progress (e.g., the objective function is being satisfactorily reduced), either increase the iterations limit and rerun nag_opt_lsq_lincon_solve (e04nc) or, alternatively, rerun nag_opt_lsq_lincon_solve (e04nc) using the Warm Start facility to specify the initial working set. If the iteration limit is already large, but some of the constraints could be nearly linearly dependent, check the monitoring information (see Section [Description of Monitoring Information]) for a repeated pattern of constraints entering and leaving the working set. (Near-dependencies are often indicated by wide variations in size in the diagonal elements of the matrix TT (see Section [Definition of Search Direction]), which will be printed if Print Level30Print Level30 (default value = 10default value=10). In this case, the algorithm could be cycling (see the comments for ifail = 5ifail=5).
  ifail = 5ifail=5
The algorithm could be cycling, since a total of 5050 changes were made to the working set without altering xx. You should check the monitoring information (see Section [Description of Monitoring Information]) for a repeated pattern of constraint deletions and additions.
If a sequence of constraint changes is being repeated, the iterates are probably cycling. (nag_opt_lsq_lincon_solve (e04nc) does not contain a method that is guaranteed to avoid cycling; such a method would be combinatorial in nature.) Cycling may occur in two circumstances: at a constrained stationary point where there are some small or zero Lagrange multipliers; or at a point (usually a vertex) where the constraints that are satisfied exactly are nearly linearly dependent. In the latter case, you have the option of identifying the offending dependent constraints and removing them from the problem, or restarting the run with a larger value of the optional parameter Feasibility Tolerance (default value = sqrt(ε)default value=ε, where εε is the machine precision). If nag_opt_lsq_lincon_solve (e04nc) terminates with ifail = 5ifail=5, but no suspicious pattern of constraint changes can be observed, it may be worthwhile to restart with the final xx (with or without the Warm Start option).
Note:  that this error exit may also occur if a poor starting point x is supplied (for example, x = 0.0x=0.0). You are advised to try a non-zero starting point.
  ifail = 6ifail=6
An input parameter is invalid.
  ifail = 7ifail=7
The problem to be solved is of type QP1 or QP2, but the Hessian matrix supplied in a is not positive semidefinite.
  OverflowOverflow
If the printed output before the overflow error contains a warning about serious ill-conditioning in the working set when adding the jjth constraint, it may be possible to avoid the difficulty by increasing the magnitude of the Feasibility Tolerance (default value = sqrt(ε)default value=ε, where εε is the machine precision) and rerunning the program. If the message recurs even after this change, the offending linearly dependent constraint (with index ‘jj’) must be removed from the problem.

Accuracy

nag_opt_lsq_lincon_solve (e04nc) implements a numerically stable active set strategy and returns solutions that are as accurate as the condition of the problem warrants on the machine.

Further Comments

This section contains some comments on scaling and a description of the printed output.

Scaling

Sensible scaling of the problem is likely to reduce the number of iterations required and make the problem less sensitive to perturbations in the data, thus improving the condition of the problem. In the absence of better information it is usually sensible to make the Euclidean lengths of each constraint of comparable magnitude. See the E04 Chapter Introduction and Gill et al. (1981) for further information and advice.

Description of the Printed Output

This section describes the intermediate printout and final printout produced by nag_opt_lsq_lincon_solve (e04nc). The intermediate printout is a subset of the monitoring information produced by the function at every iteration (see Section [Description of Monitoring Information]). You can control the level of printed output (see the description of the optional parameter Print Level). Note that the intermediate printout and final printout are produced only if Print Level10Print Level10 (the default for nag_opt_lsq_lincon_solve (e04nc), by default no output is produced by nag_opt_lsq_lincon_solve (e04nc)).
The following line of summary output ( < 80<80 characters) is produced at every iteration. In all cases, the values of the quantities printed are those in effect on completion of the given iteration.
Itn is the iteration count.
Step is the step taken along the computed search direction. If a constraint is added during the current iteration (i.e., Jadd is positive), Step will be the step to the nearest constraint. During the optimality phase, the step can be greater than one only if the factor RZRZ is singular. (See Section [Main Iteration].)
Ninf is the number of violated constraints (infeasibilities). This will be zero during the optimality phase.
Sinf/Objective is the value of the current objective function. If xx is not feasible, Sinf gives a weighted sum of the magnitudes of constraint violations. If xx is feasible, Objective is the value of the objective function of (1). The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point.
During the optimality phase the value of the objective function will be nonincreasing. During the feasibility phase the number of constraint infeasibilities will not increase until either a feasible point is found or the optimality of the multipliers implies that no feasible point exists. Once optimal multipliers are obtained the number of infeasibilities can increase, but the sum of infeasibilities will either remain constant or be reduced until the minimum sum of infeasibilities is found.
Norm Gz is Z1TgFR Z1T gFR , the Euclidean norm of the reduced gradient with respect to Z1Z1. During the optimality phase, this norm will be approximately zero after a unit step. (See Sections [Definition of Search Direction] and [Main Iteration].)
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 general constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, bl(j)blj and bu(j)buj are replaced by bl(n + j)bln+j and bu(n + j)bun+j respectively, and with the following change in the heading:
L Con gives the name (L) and index jj, for j = 1,2,,nLj=1,2,,nL, of the linear 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_lsq_lincon_solve_example
c = [1, 1, 1, 1, 1, 1, 1, 1, 4;
     1, 2, 3, 4, -2, 1, 1, 1, 1;
     1, -1, 1, -1, 1, 1, 1, 1, 1];
bl = [0;
     0;
     -1e25;
     0;
     0;
     0;
     0;
     0;
     0;
     2;
     -1e25;
     1];
bu = [2;
     2;
     2;
     2;
     2;
     2;
     2;
     2;
     2;
     1e25;
     2;
     4];
cvec = [0];
istate= zeros(12, 1, 'int64');
kx = zeros(9, 1, 'int64');
x = [1;
     0.5;
     0.3333;
     0.25;
     0.2;
     0.1667;
     0.1428;
     0.125;
     0.1111];
a = [1,   1,   1,   1,   1,   1,   1,   1,   1;
     1,   2,   1,   1,   1,   1,   2,   0,   0;
     1,   1,   3,   1,   1,   1,  -1,  -1,  -3;
     1,   1,   1,   4,   1,   1,   1,   1,   1;
     1,   1,   1,   3,   1,   1,   1,   1,   1;
     1,   1,   2,   1,   1,   0,   0,   0,  -1;
     1,   1,   1,   1,   0,   1,   1,   1,   1;
     1,   1,   1,   0,   1,   1,   1,   1,   1;
     1,   1,   0,   1,   1,   1,   2,   2,   3;
     1,   0,   1,   1,   1,   1,   0,   2,   2 ];
b = [1,   1,   1,   1,   1,   1,   1,   1,   1,   1,];
[cwsav,lwsav,iwsav,rwsav,ifail] = nag_opt_init('nag_opt_lsq_lincon_solve');
[istateOut, kxOut, xOut, aOut, bOut, iter, obj, clamda, lwsavOut, iwsavOut, rwsavOut, ifail] = ...
    nag_opt_lsq_lincon_solve(c, bl, bu, cvec, istate, kx, x, a, b, lwsav, iwsav, rwsav);
 istateOut, kxOut, xOut, aOut, bOut, iter, obj, clamda, ifail
 

istateOut =

                    1
                    0
                    0
                    1
                    0
                    1
                    0
                    1
                    0
                    1
                    2
                    1


kxOut =

                    3
                    9
                    2
                    5
                    7
                    6
                    8
                    1
                    4


xOut =

         0
    0.0415
    0.5872
         0
    0.0996
         0
    0.0491
         0
    0.3056


aOut =

    4.8017    0.2202    3.1226    0.4718    1.7539    2.1469    0.5193    2.5809    3.5110
    0.1502    3.0381    0.3365   -3.4674   -2.1886   -1.0342   -1.6942   -0.8582   -1.4536
    0.1502    0.6322   -1.0175    1.0208    1.2471    0.1862    1.0728    0.2052    0.7820
    0.6009    0.1227   -0.3962   -3.0130   -2.8247   -1.5832   -3.1160   -1.6001   -2.2064
    0.4507    0.0458   -0.2083    0.1533    0.0000    0.7523    0.0000   -0.0000   -1.5022
    0.1502    0.2620    0.2116    0.0654   -0.0538    0.4627    0.0000    0.0000    3.1342
    0.1502   -0.1081    0.1676   -0.1745    0.9495    0.0505   -0.0000   -0.0000    0.0000
         0   -0.1851    0.3556   -0.3385   -0.1728   -0.0329   -0.1766    0.0000   -0.0000
    0.1502   -0.4783    0.1236   -0.4145   -0.1743    0.0118   -0.4615    0.2966   -0.0000
    0.1502   -0.2932    0.2715    0.1445   -0.1684   -0.0106    0.1221   -0.2072    0.4538


bOut =

  Columns 1 through 9

   -0.0000         0   -0.2218    0.3369   -0.0000    0.0000    0.0000   -0.0000   -0.0000

  Column 10

    0.0000


iter =

                   12


obj =

    0.0813


clamda =

    0.1572
         0
         0
    0.8782
         0
    0.1473
         0
    0.8603
         0
    0.3777
   -0.0579
    0.1075


ifail =

                    0


function e04nc_example
c = [1, 1, 1, 1, 1, 1, 1, 1, 4;
     1, 2, 3, 4, -2, 1, 1, 1, 1;
     1, -1, 1, -1, 1, 1, 1, 1, 1];
bl = [0;
     0;
     -1e25;
     0;
     0;
     0;
     0;
     0;
     0;
     2;
     -1e25;
     1];
bu = [2;
     2;
     2;
     2;
     2;
     2;
     2;
     2;
     2;
     1e25;
     2;
     4];
cvec = [0];
istate= zeros(12, 1, 'int64');
kx = zeros(9, 1, 'int64');
x = [1;
     0.5;
     0.3333;
     0.25;
     0.2;
     0.1667;
     0.1428;
     0.125;
     0.1111];
a = [1,   1,   1,   1,   1,   1,   1,   1,   1;
     1,   2,   1,   1,   1,   1,   2,   0,   0;
     1,   1,   3,   1,   1,   1,  -1,  -1,  -3;
     1,   1,   1,   4,   1,   1,   1,   1,   1;
     1,   1,   1,   3,   1,   1,   1,   1,   1;
     1,   1,   2,   1,   1,   0,   0,   0,  -1;
     1,   1,   1,   1,   0,   1,   1,   1,   1;
     1,   1,   1,   0,   1,   1,   1,   1,   1;
     1,   1,   0,   1,   1,   1,   2,   2,   3;
     1,   0,   1,   1,   1,   1,   0,   2,   2 ];
b = [1,   1,   1,   1,   1,   1,   1,   1,   1,   1,];
[cwsav,lwsav,iwsav,rwsav,ifail] = e04wb('e04nc');
[istateOut, kxOut, xOut, aOut, bOut, iter, obj, clamda, lwsavOut, iwsavOut, rwsavOut, ifail] = ...
    e04nc(c, bl, bu, cvec, istate, kx, x, a, b, lwsav, iwsav, rwsav);
 istateOut, kxOut, xOut, aOut, bOut, iter, obj, clamda, ifail
 

istateOut =

                    1
                    0
                    0
                    1
                    0
                    1
                    0
                    1
                    0
                    1
                    2
                    1


kxOut =

                    3
                    9
                    2
                    5
                    7
                    6
                    8
                    1
                    4


xOut =

         0
    0.0415
    0.5872
         0
    0.0996
         0
    0.0491
         0
    0.3056


aOut =

    4.8017    0.2202    3.1226    0.4718    1.7539    2.1469    0.5193    2.5809    3.5110
    0.1502    3.0381    0.3365   -3.4674   -2.1886   -1.0342   -1.6942   -0.8582   -1.4536
    0.1502    0.6322   -1.0175    1.0208    1.2471    0.1862    1.0728    0.2052    0.7820
    0.6009    0.1227   -0.3962   -3.0130   -2.8247   -1.5832   -3.1160   -1.6001   -2.2064
    0.4507    0.0458   -0.2083    0.1533    0.0000    0.7523    0.0000   -0.0000   -1.5022
    0.1502    0.2620    0.2116    0.0654   -0.0538    0.4627    0.0000    0.0000    3.1342
    0.1502   -0.1081    0.1676   -0.1745    0.9495    0.0505   -0.0000   -0.0000    0.0000
         0   -0.1851    0.3556   -0.3385   -0.1728   -0.0329   -0.1766    0.0000   -0.0000
    0.1502   -0.4783    0.1236   -0.4145   -0.1743    0.0118   -0.4615    0.2966   -0.0000
    0.1502   -0.2932    0.2715    0.1445   -0.1684   -0.0106    0.1221   -0.2072    0.4538


bOut =

  Columns 1 through 9

   -0.0000         0   -0.2218    0.3369   -0.0000    0.0000    0.0000   -0.0000   -0.0000

  Column 10

    0.0000


iter =

                   12


obj =

    0.0813


clamda =

    0.1572
         0
         0
    0.8782
         0
    0.1473
         0
    0.8603
         0
    0.3777
   -0.0579
    0.1075


ifail =

                    0


Note: 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_lsq_lincon_option_string (e04ne). 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_lsq_lincon_solve (e04nc).

Overview

nag_opt_lsq_lincon_solve (e04nc) is essentially identical to the function LSSOL described in Gill et al. (1986). It is based on a two-phase (primal) quadratic programming method with features to exploit the convexity of the objective function due to Gill et al. (1984). (In the full-rank case, the method is related to that of Stoer (1971).) nag_opt_lsq_lincon_solve (e04nc) has two phases: 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 two-phase nature of the algorithm is reflected by changing the function being minimized from the sum of infeasibilities to the quadratic objective function. The feasibility phase does not perform the standard simplex method (i.e., it does not necessarily find a vertex), except in the LP case when nLnnLn. Once any iterate is feasible, all subsequent iterates remain feasible.
nag_opt_lsq_lincon_solve (e04nc) has been designed to be efficient when used to solve a sequence of related problems – for example, within a sequential quadratic programming method for nonlinearly constrained optimization (e.g., nag_opt_nlp1_rcomm (e04uf) or nag_opt_nlp2_solve (e04wd)). In particular, you may specify an initial working set (the indices of the constraints believed to be satisfied exactly at the solution); see the discussion of the optional parameter Warm Start.
In general, an iterative process is required to solve a quadratic program. (For simplicity, we shall always consider a typical iteration and avoid reference to the index of the iteration.) Each new iterate xx- is defined by
x = x + αp,
x-=x+αp,
(2)
where the step length αα is a non-negative scalar, and pp is called the search direction.
At each point xx, a working set of constraints is defined to be a linearly independent subset of the constraints that are satisfied ‘exactly’ (to within the tolerance defined by the optional parameter Feasibility Tolerance). The working set is the current prediction of the constraints that hold with equality at a solution of (1). The search direction is constructed so that the constraints in the working set remain unaltered for any value of the step length. For a bound constraint in the working set, this property is achieved by setting the corresponding element of the search direction to zero. Thus, the associated variable is fixed, and specification of the working set induces a partition of xx into fixed and free variables. During a given iteration, the fixed variables are effectively removed from the problem; since the relevant elements of the search direction are zero, the columns of cc corresponding to fixed variables may be ignored.
Let nWnW denote the number of general constraints in the working set and let nFXnFX denote the number of variables fixed at one of their bounds (nWnW and nFXnFX are the quantities Lin and Bnd in the monitoring file output from nag_opt_lsq_lincon_solve (e04nc); see Section [Description of Monitoring Information]). Similarly, let nFR(nFR = nnFX)nFR(nFR=n-nFX) denote the number of free variables. At every iteration, the variables are reordered so that the last nFXnFX variables are fixed, with all other relevant vectors and matrices ordered accordingly. The order of the variables is indicated by the contents of the array kx on exit (see Section [Parameters]).

Definition of Search Direction

Let CFRCFR denote the nWnW by nFRnFR sub-matrix of general constraints in the working set corresponding to the free variables, and let pFRpFR denote the search direction with respect to the free variables only. The general constraints in the working set will be unaltered by any move along pp if
CFRpFR = 0.
CFRpFR=0.
(3)
In order to compute pFRpFR, the TQTQ factorization of CFRCFR is used:
CFRQFR = (0T)
CFRQFR=(0T)
(4)
where TT is a nonsingular nWnW by nWnW reverse-triangular matrix (i.e., tij = 0tij=0 if i + j < nWi+j<nW), and the nonsingular nFRnFR by nFRnFR matrix QFRQFR is the product of orthogonal transformations (see Gill et al. (1984)). If the columns of QFRQFR are partitioned so that
QFR = (ZY),
QFR=(ZY),
(5)
where YY is nFRnFR by nWnW, then the nZ(nZ = nFRnW)nZ(nZ=nFR-nW) columns of ZZ form a basis for the null space of CFRCFR. Let nRnR be an integer such that 0nRnZ0nRnZ, and let Z1Z1 denote a matrix whose nRnR columns are a subset of the columns of ZZ. (The integer nRnR is the quantity Zr in the monitoring file output from nag_opt_lsq_lincon_solve (e04nc). In many cases, Z1Z1 will include all the columns of ZZ.) The direction pFRpFR will satisfy (3) if
pFR = Z1pZ
pFR=Z1pZ
(6)
where pZpZ is any nRnR-vector.

Main Iteration

Let QQ denote the nn by nn matrix
Q =
(QFR)
IFX
,
Q= QFR IFX ,
(7)
where IFXIFX is the identity matrix of order nFXnFX. Let RR denote an nn by nn upper triangular matrix (the Cholesky factor) such that
RTR = HQQTQ,
RTR=HQQTH~Q,
(8)
where H~ is the Hessian HH with rows and columns permuted so that the free variables are first.
Let the matrix of the first nZnZ rows and columns of RR be denoted by RZRZ. The definition of pZpZ in (6) depends on whether or not the matrix RZRZ is singular at xx. In the nonsingular case, pZpZ satisfies the equations
RZT RZ pZ = gZ
RZT RZ pZ = - gZ
(9)
where gZgZ denotes the vector ZTgFRZTgFR and gg denotes the objective gradient. (The norm of gFRgFR is the printed quantity Norm Gf; see Section [Description of Monitoring Information].) When pZpZ is defined by (9), x + px+p is the minimizer of the objective function subject to the constraints (bounds and general) in the working set treated as equalities. In general, a vector fZfZ is available such that RZT fZ = gZ RZT fZ = - gZ , which allows pZpZ to be computed from a single back-substitution RZpZ = fZRZpZ=fZ. For example, when solving problem LS1, fZfZ comprises the first nZnZ elements of the transformed residual vector
f = P(bAx),
f=P(b-Ax),
(10)
which is recurred from one iteration to the next, where PP is an orthogonal matrix.
In the singular case, pZpZ is defined such that
RZ pZ = 0   and   gZT pZ < 0 .
RZ pZ = 0   and   gZT pZ < 0 .
(11)
This vector has the property that the objective function is linear along pp and may be reduced by any step of the form x + αpx+αp, where α > 0α>0.
The vector ZTgFRZTgFR is known as the projected gradient at xx. If the projected gradient is zero, xx is a constrained stationary point in the subspace defined by ZZ. During the feasibility phase, the projected gradient will usually be zero only at a vertex (although it may be zero at non-vertices in the presence of constraint dependencies). During the optimality phase, a zero projected gradient implies that xx minimizes the quadratic objective when the constraints in the working set are treated as equalities. At a constrained stationary point, Lagrange multipliers λcλc and λbλb for the general and bound constraints are defined from the equations
CFRTλC = gFRand ​λB = gFXCFXTλC.
CFRTλC=gFRand ​λB=gFX-CFXTλC.
(12)
Given a positive constant δδ of the order of the machine precision, the Lagrange multiplier λjλj corresponding to an inequality constraint in the working set is said to be optimal if λjδλjδ when the associated constraint is at its upper bound, or if λjδλj-δ when the associated constraint is at its lower bound. If a multiplier is nonoptimal, the objective function (either the true objective or the sum of infeasibilities) can be reduced by deleting the corresponding constraint (with index Jdel; see Section [Description of Monitoring Information]) from the working set.
If optimal multipliers occur during the feasibility phase and the sum of infeasibilities is nonzero, there is no feasible point, and nag_opt_lsq_lincon_solve (e04nc) will continue until the minimum value of the sum of infeasibilities has been found. At this point, the Lagrange multiplier λjλj corresponding to an inequality constraint in the working set will be such that (1 + δ)λjδ-(1+δ)λjδ when the associated constraint is at its upper bound, and δλj(1 + δ)-δλj(1+δ) when the associated constraint is at its lower bound. Lagrange multipliers for equality constraints will satisfy |λj|1 + δ|λj|1+δ.
The choice of step length is based on remaining feasible with respect to the satisfied constraints. If RZRZ is nonsingular and x + px+p is feasible, αα will be taken as unity. In this case, the projected gradient at xx- will be zero, and Lagrange multipliers are computed. Otherwise, αα is set to αmαm, the step to the ‘nearest’ constraint (with index Jadd; see Section [Description of Monitoring Information]), which is added to the working set at the next iteration.
If AA is not input as a triangular matrix, it is overwritten by a triangular matrix RR satisfying (8) obtained using the Cholesky factorization in the QP case, or the QRQR factorization in the LS case. Column interchanges are used in both cases, and an estimate is made of the rank of the triangular factor. Thereafter, the dependent rows of RR are eliminated from the problem.
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 T,QFRT,QFR and RR; and of vectors QTgQTg, QTcQTc and ff, which are related by the formulae
f = Pb
(R)
0
QTx ,   (b0for the ​QP​ case) ,
f=Pb- R 0 QTx ,   ( b0 for the ​ QP ​ case ) ,
and
QTg = QTcRTf.
QTg=QTc-RTf.
Note that the triangular factor RR associated with the Hessian of the original problem is updated during both the optimality and the feasibility phases.
The treatment of the singular case depends critically on the following feature of the matrix updating schemes used in nag_opt_lsq_lincon_solve (e04nc): if a given factor RZRZ is nonsingular, it can become singular during subsequent iterations only when a constraint leaves the working set, in which case only its last diagonal element can become zero. This property implies that a vector satisfying (11) may be found using the single back-substitution RZpZ = eZR-ZpZ=eZ, where RZR-Z is the matrix RZRZ with a unit last diagonal, and eZeZ is a vector of all zeros except in the last position. If HH is singular, the matrix RR (and hence RZRZ) may be singular at the start of the optimality phase. However, RZRZ will be nonsingular if enough constraints are included in the initial working set. (The matrix with no rows and columns is positive definite by definition, corresponding to the case when CFRCFR contains nFRnFR constraints.) The idea is to include as many general constraints as necessary to ensure a nonsingular RZRZ.
At the beginning of each phase, an upper triangular matrix R1R1 is determined that is the largest nonsingular leading sub-matrix of RZRZ. The use of interchanges during the factorization of AA tends to maximize the dimension of R1R1. (The rank of R1R1 is estimated using the optional parameter Rank Tolerance.) Let Z1Z1 denote the columns of ZZ corresponding to R1R1, and let ZZ be partitioned as Z = (Z1Z2)Z=(Z1Z2). A working set for which Z1Z1 defines the null space can be obtained by including the rows of Z2T Z2T  as ‘artificial constraints’. Minimization of the objective function then proceeds within the subspace defined by Z1Z1.
The artificially augmented working set is given by
CFR =
(CFR)
Z2T
,
C-FR= CFR Z2T ,
(13)
so that pFRpFR will satisfy CFRpFR = 0CFRpFR=0 and Z2T pFR = 0 Z2T pFR = 0 . By definition of the TQTQ factorization, cFRc-FR automatically satisfies the following:
CFRQFR =
(CFR)
Z2T
QFR =
(CFR)
Z2T
(Z1Z2Y)
=
(0T)
,
C-FRQFR= CFR Z2T QFR= CFR Z2T Z1 Z2 Y = 0 T- ,
where
T =
(0T)
I 0
,
T-= 0 T I 0 ,
and hence the TQTQ factorization of (13) requires no additional work.
The matrix Z2Z2 need not be kept fixed, since its role is purely to define an appropriate null space; the TQTQ factorization can therefore be updated in the normal fashion as the iterations proceed. No work is required to ‘delete’ the artificial constraints associated with Z2Z2 when Z1T gFR = 0 Z1T gFR = 0 , since this simply involves repartitioning QFRQFR. When deciding which constraint to delete, the ‘artificial’ multiplier vector associated with the rows of Z2T Z2T  is equal to Z2T gFR Z2T gFR , and the multipliers corresponding to the rows of the ‘true’ working set are the multipliers that would be obtained if the temporary constraints were not present.
The number of columns in Z2Z2 and Z1 Z1 , the Euclidean norm of Z1T gFR Z1T gFR , and the condition estimator of R1R1 appear in the monitoring file output as Art, Zr, Norm Gz and Cond Rz respectively (see Section [Description of Monitoring Information]).
Although the algorithm of nag_opt_lsq_lincon_solve (e04nc) does not perform simplex steps in general, there is one exception: a linear program with fewer general constraints than variables (i.e., nLnnLn). Use of the simplex method in this situation leads to savings in storage. At the starting point, the ‘natural’ working set (the set of constraints exactly or nearly satisfied at the starting point) is augmented with a suitable number of ‘temporary’ bounds, each of which has the effect of temporarily fixing a variable at its current value. In subsequent iterations, a temporary bound is treated as a standard constraint until it is deleted from the working set, in which case it is never added again.
One of the most important features of nag_opt_lsq_lincon_solve (e04nc) is its control of the conditioning of the working set, whose nearness to linear dependence is estimated by the ratio of the largest to smallest diagonals of the TQTQ factor TT (the printed value Cond T; see Section [Description of Monitoring Information]). In constructing the initial working set, constraints are excluded that would result in a large value of Cond T. Thereafter, nag_opt_lsq_lincon_solve (e04nc) allows constraints to be violated by as much as a user-specified optional parameter Feasibility Tolerance in order to provide, whenever possible, a choice of constraints to be added to the working set at a given iteration. Let αmαm denote the maximum step at which x + αmpx+αmp does not violate any constraint by more than its feasibility tolerance. All constraints at distance α(ααm)α(ααm) along pp from the current point are then viewed as acceptable candidates for inclusion in the working set. The constraint whose normal makes the largest angle with the search direction is added to the working set. In order to ensure that the new iterate satisfies the constraints in the working set as accurately as possible, the step taken is the exact distance to the newly added constraint. As a consequence, negative steps are occasionally permitted, since the current iterate may violate the constraint to be added by as much as the feasibility tolerance.

Optional Parameters

Several optional parameters in nag_opt_lsq_lincon_solve (e04nc) define choices in the problem specification or the algorithm logic. In order to reduce the number of formal parameters of nag_opt_lsq_lincon_solve (e04nc) 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_lsq_lincon_option_string (e04ne) before a call to nag_opt_lsq_lincon_solve (e04nc).
nag_opt_lsq_lincon_option_string (e04ne) can be called to supply options directly, one call being necessary for each optional parameter. For example,
[lwsav, iwsav, rwsav, inform] = e04ne('Print Level = 1', lwsav, iwsav, rwsav);
nag_opt_lsq_lincon_option_string (e04ne) should be consulted for a full description of this method of supplying optional parameters.
All optional parameters not specified by you are set to their default values. Optional parameters specified by you are unaltered by nag_opt_lsq_lincon_solve (e04nc) (unless they define invalid values) and so remain in effect for subsequent calls unless altered by you.

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.
Cold Start  
Default
Warm Start  
This option specifies how the initial working set is chosen. With a Cold Start, nag_opt_lsq_lincon_solve (e04nc) chooses the initial working set 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 provide a valid definition of every element of the array istate. nag_opt_lsq_lincon_solve (e04nc) 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_lsq_lincon_solve (e04nc) 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_lsq_lincon_solve (e04nc) 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 cjT xl cjT xl  will be included in the initial working set if |cjTxl| r (1 + |l|) | cjT 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.
Feasibility Phase Iteration Limit  i1i1
Default = max (50,5(n + nL))=max(50,5(n+nL))
Optimality Phase Iteration Limit  i2i2
Default = max (50,5(n + nL))=max(50,5(n+nL))
The scalars i1i1 and i2i2 specify the maximum number of iterations allowed in the feasibility and optimality phases. Optional parameter Optimality Phase Iteration Limit is equivalent to optional parameter Iteration Limit. Setting i2 = 0i2=0 and Print Level > 0Print Level>0 means that the workspace needed will be computed and printed, but no iterations will be performed. If i1 < 0i1<0 or i2 < 0i2<0, the default value is used.
Feasibility Tolerance  rr
Default = sqrt(ε)=ε
If r > εr>ε, rr defines the maximum acceptable absolute violation in each constraint at a ‘feasible’ point. For example, if the variables and the coefficients in the general constaints are of order unity, and the latter are correct to about 66 decimal digits, it would be appropriate to specify rr as 10610-6. If 0r < ε0r<ε, the default value is used.
Note that a ‘feasible solution’ is a solution that satisfies the current constraints to within the tolerance rr.
Hessian  NoNo
Default = NO=NO 
This option controls the contents of the upper triangular matrix RR (see the description of a in Section [Parameters]). nag_opt_lsq_lincon_solve (e04nc) works exclusively with the transformed and reordered matrix HQHQ (8), and hence extra computation is required to form the Hessian itself. If Hessian = NOHessian=NO, a contains the Cholesky factor of the matrix HQHQ with columns ordered as indicated by kx (see Section [Parameters]). If Hessian = YESHessian=YES, a contains the Cholesky factor of the matrix HH, with columns ordered as indicated by kx.
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 will be considered a step to an unbounded solution. (Note that an unbounded solution can occur only when the Hessian is singular and the objective contains an explicit linear term.) 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.
Iteration Limit  ii
Default = max (50,5(n + nL))=max(50,5(n+nL))
Iters  
Itns  
See optional parameter Feasibility Phase Iteration Limit.
List  
Default for e04nc = Liste04nc=List
Nolist  
Default for e04nc = Noliste04nc=Nolist
Normally each optional parameter specification is printed as it is supplied. Optional parameter Nolist may be used to suppress the printing and optional parameter List may be used to restore printing.
Monitoring File  ii
Default = 1=-1
If i0i0 and Print Level5Print Level5, monitoring information produced by nag_opt_lsq_lincon_solve (e04nc) at every iteration is sent to a file with logical unit number ii. If i < 0i<0 and/or Print Level < 5Print Level<5, no monitoring information is produced.
Print Level  ii
The value of ii controls the amount of printout produced by nag_opt_lsq_lincon_solve (e04nc), as indicated below. A detailed description of the printed output is given in Section [Printed output] (summary output at each iteration and the final solution) and Section [Description of Monitoring Information] (monitoring information at each iteration).
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 iteration (no printout of the final solution).
1010 The final solution and one line of summary output for each 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 [Description of Monitoring Information]) for each iteration (no printout of the final solution).
2020 At each iteration, the Lagrange multipliers, the variables xx, the constraint values CxCx and the constraint status.
3030 At each iteration, the diagonal elements of the matrix TT associated with the TQTQ factorization (4) (see Section [Definition of Search Direction]) of the working set, and the diagonal elements of the upper triangular matrix RR.
If Print Level5Print 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 is suppressed.
Problem Type  aa
Default = = LS1
This option specifies the type of objective function to be minimized during the optimality phase. The following are the nine optional keywords and the dimensions of the arrays that must be specified in order to define the objective function:
LP a and b not referenced, cvec(n)cvecn;
QP1 a(lda,n)aldan symmetric, b and cvec not referenced;
QP2 a(lda,n)aldan symmetric, b not referenced, cvec(n)cvecn;
QP3 a(lda,n)aldan upper trapezoidal, kx(n)kxn, b and cvec not referenced;
QP4 a(lda,n)aldan upper trapezoidal, kx(n)kxn, b not referenced, cvec(n)cvecn;
LS1 a(lda,n)aldan, b(m)bm, cvec not referenced;
LS2 a(lda,n)aldan, b(m)bm, cvec(n)cvecn;
LS3 a(lda,n)aldan upper trapezoidal, kx(n)kxn, b(m)bm, cvec not referenced;
LS4 a(lda,n)aldan upper trapezoidal, kx(n)kxn, b(m)bm, cvec(n)cvecn.
For problems of type FP, the objective function is omitted and a, b and cvec are not referenced.
The following keywords are also acceptable. The minimum abbreviation of each keyword is underlined.
aa Option
Least LS1
Quadratic QP2
Linear LP
In addition, the keywords LS and LSQ are equivalent to the default option LS1, and the keyword QP is equivalent to the option QP2.
If A = 0A=0, i.e., the objective function is purely linear, the efficiency of nag_opt_lsq_lincon_solve (e04nc) may be increased by specifying aa as LP.
Rank Tolerance  rr
Default = 100ε=100ε or 10sqrt(ε)10ε (see below)
Note that this option does not apply to problems of type FP or LP.
The default value of rr depends on the problem type. If AA occurs as a least squares matrix, as it does in problem types QP1, LS1 and LS3, then the default value of rr is 100ε100ε. In all other cases, AA is treated as the ‘square root’ of the Hessian matrix HH and rr has the default value 10sqrt(ε)10ε.
This parameter enables you to control the estimate of the triangular factor R1R1 (see Section [Main Iteration]). If ρiρi denotes the function ρi = max {|R11|,|R22|,,|Rii|}ρi=max{|R11|,|R22|,,|Rii|}, the rank of RR is defined to be smallest index i such that |Ri + 1,i + 1|r|ρi + 1||Ri+1,i+1|r|ρi+1|. If r0r0, the default value is used.

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_lsq_lincon_solve (e04nc). (See also the description of the optional parameters Monitoring File and Print Level.) You can control the level of printed output.
To aid interpretation of the printed results, the following convention is used for numbering the constraints: indices 11 through nn refer to the bounds on the variables, and indices n + 1n+1 through n + nLn+nL refer to the general constraints. When the status of a constraint changes, the index of the constraint is printed, along with the designation L (lower bound), U (upper bound), E (equality), F (temporarily fixed variable) or A (artificial constraint).
When Print Level5Print Level5 and Monitoring File0Monitoring File0, the following line of output is produced at every iteration on the unit number specified by optional parameter Monitoring File. In all cases, the values of the quantities printed are those in effect on completion of the given iteration.
Itn is the iteration count.
Jdel is the index of the constraint deleted from the working set. If Jdel is zero, no constraint was deleted.
Jadd is the index of the constraint added to the working set. If Jadd is zero, no constraint was added.
Step is the step taken along the computed search direction. If a constraint is added during the current iteration (i.e., Jadd is positive), Step will be the step to the nearest constraint. During the optimality phase, the step can be greater than one only if the factor RZRZ is singular.
Ninf is the number of violated constraints (infeasibilities). This will be zero during the optimality phase.
Sinf/Objective is the value of the current objective function. If xx is not feasible, Sinf gives a weighted sum of the magnitudes of constraint violations. If xx is feasible, Objective is the value of the objective function of (1). The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point.
During the optimality phase the value of the objective function will be nonincreasing. During the feasibility phase the number of constraint infeasibilities will not increase until either a feasible point is found or the optimality of the multipliers implies that no feasible point exists. Once optimal multipliers are obtained the number of infeasibilities can increase, but the sum of infeasibilities will either remain constant or be reduced until the minimum sum of infeasibilities is found.
Bnd is the number of simple bound constraints in the current working set.
Lin is the number of general linear constraints in the current working set.
Art is the number of artificial constraints in the working set, i.e., the number of columns of Z2Z2 (see Section [Main Iteration]).
Zr is the number of columns of Z1Z1(see Section [Definition of Search Direction]). Zr is the dimension of the subspace in which the objective function is currently being minimized. The value of Zr is the number of variables minus the number of constraints in the working set; i.e., Zr = n(Bnd + Lin + Art)Zr=n-(Bnd+Lin+Art).
The value of nZnZ, the number of columns of ZZ (see Section [Definition of Search Direction]) can be calculated as nZ = n(Bnd + Lin)nZ=n-(Bnd+Lin). A zero value of nZnZ implies that xx lies at a vertex of the feasible region.
Norm Gz is Z1TgFR Z1T gFR , the Euclidean norm of the reduced gradient with respect to Z1Z1. During the optimality phase, this norm will be approximately zero after a unit step.
Norm Gf is the Euclidean norm of the gradient function with respect to the free variables, i.e., variables not currently held at a bound.
Cond T is a lower bound on the condition number of the working set.
Cond Rz is a lower bound on the condition number of the triangular factor R1R1 (the first Zr rows and columns of the factor RZRZ). If the problem is specified to be of type LP or the estimated rank of the data matrix AA is zero then Cond Rz is not printed.

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013