NAG Library Routine Document
E04UFF/E04UFA
Note: this routine uses optional parameters to define choices in the problem specification and in the details of the algorithm. If you wish to use default
settings for all of the optional parameters, you need only read Sections 1 to 9 of this document. If, however, you wish to reset some or all of the settings please refer to Section 10 for a detailed description of the algorithm, to Section 11 for a detailed description of the specification of the optional parameters and to Section 12 for a detailed description of the monitoring information produced by the routine.
1 Purpose
E04UFF/E04UFA 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.
E04UFF/E04UFA may also be used for unconstrained, boundconstrained and linearly constrained optimization.
E04UFF/E04UFA uses reverse communication for evaluating the objective function, the nonlinear constraint functions and any of their derivatives.
E04UFA is a version of E04UFF that has additional parameters in order to make it safe for use in multithreaded applications (see
Section 5). The initialization routine
E04WBF must have been called before calling E04UFA.
2 Specification
2.1 Specification for E04UFF
SUBROUTINE E04UFF ( 
IREVCM, N, NCLIN, NCNLN, LDA, LDCJ, LDR, A, BL, BU, ITER, ISTATE, C, CJAC, CLAMDA, OBJF, OBJGRD, R, X, NEEDC, IWORK, LIWORK, WORK, LWORK, IFAIL) 
INTEGER 
IREVCM, N, NCLIN, NCNLN, LDA, LDCJ, LDR, ITER, ISTATE(N+NCLIN+NCNLN), NEEDC(max(1,NCNLN)), IWORK(LIWORK), LIWORK, LWORK, IFAIL 
REAL (KIND=nag_wp) 
A(LDA,*), BL(N+NCLIN+NCNLN), BU(N+NCLIN+NCNLN), C(*), CJAC(LDCJ,*), CLAMDA(N+NCLIN+NCNLN), OBJF, OBJGRD(N), R(LDR,N), X(N), WORK(LWORK) 

2.2 Specification for E04UFA
SUBROUTINE E04UFA ( 
IREVCM, N, NCLIN, NCNLN, LDA, LDCJ, LDR, A, BL, BU, ITER, ISTATE, C, CJAC, CLAMDA, OBJF, OBJGRD, R, X, NEEDC, IWORK, LIWORK, WORK, LWORK, CWSAV, LWSAV, IWSAV, RWSAV, IFAIL) 
INTEGER 
IREVCM, N, NCLIN, NCNLN, LDA, LDCJ, LDR, ITER, ISTATE(N+NCLIN+NCNLN), NEEDC(max(1,NCNLN)), IWORK(LIWORK), LIWORK, LWORK, IWSAV(610), IFAIL 
REAL (KIND=nag_wp) 
A(LDA,*), BL(N+NCLIN+NCNLN), BU(N+NCLIN+NCNLN), C(*), CJAC(LDCJ,*), CLAMDA(N+NCLIN+NCNLN), OBJF, OBJGRD(N), R(LDR,N), X(N), WORK(LWORK), RWSAV(475) 
LOGICAL 
LWSAV(120) 
CHARACTER(80) 
CWSAV(5) 

Before calling E04UFA, or either of the option setting routines
E04UDA or
E04UEA,
E04WBF must be called. The specification for
E04WBF is:
INTEGER 
LCWSAV, LLWSAV, IWSAV(LIWSAV), LIWSAV, LRWSAV, IFAIL 
REAL (KIND=nag_wp) 
RWSAV(LRWSAV) 
LOGICAL 
LWSAV(LLWSAV) 
CHARACTER(*) 
RNAME 
CHARACTER(80) 
CWSAV(LCWSAV) 

E04WBF should be called with
${\mathbf{RNAME}}=\text{'E04UFA'}$.
CWSAV must have a length of
LCWSAV
while the lengths of
LWSAV,
IWSAV and
RWSAV should be
LLWSAV,
LIWSAV and
LRWSAV, respectively. These parameters must satisfy the following constraints:
 ${\mathbf{LCWSAV}}\ge 5$
 ${\mathbf{LLWSAV}}\ge 120$
 ${\mathbf{LIWSAV}}\ge 610$
 ${\mathbf{LRWSAV}}\ge 475$
The contents of the arrays
CWSAV,
LWSAV,
IWSAV and
RWSAV
must not be altered between calling routines
E04UDA,
E04UEA,
E04UFA or
E04WBF.
3 Description
E04UFF/E04UFA 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:
where
$F\left(x\right)$ (the
objective function) is a nonlinear function,
${A}_{L}$ is an
${n}_{L}$ by
$n$ constant matrix, and
$c\left(x\right)$ is an
${n}_{N}$ element vector of nonlinear constraint functions. (The matrix
${A}_{L}$ and the vector
$c\left(x\right)$ may be empty.) The objective function and the constraint functions are assumed to be smooth, i.e., at least twicecontinuously differentiable. (The method of E04UFF/E04UFA 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
${l}_{i}={u}_{i}$. If certain bounds are not present, the associated elements of
$l$ or
$u$ can be set to special values that will be treated as
$\infty $ or
$+\infty $. (See the description of the optional parameter
Infinite Bound Size.)
If there are no nonlinear constraints in
(1) and
$F$ is linear or quadratic then it will generally be more efficient to use one of
E04MFF/E04MFA,
E04NCF/E04NCA or
E04NFF/E04NFA, or
E04NQF if the problem is large and sparse. If the problem is large and sparse and does have nonlinear constraints,
E04UGF/E04UGA should be used, since E04UFF/E04UFA treats all matrices as dense.
E04UFF/E04UFA uses reverse communication for evaluating
$F\left(x\right)$,
$c\left(x\right)$ 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 reenter the routine.
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 E04UFF/E04UFA is described in detail in
Section 10.
E04WDF is an alternative routine which uses a similar method, but with
forward communication:
that is, the objective and constraint functions are evaluated by subroutines, supplied as parameters to the routine.
4 References
Dennis J E Jr and Moré J J (1977) QuasiNewton methods, motivation and theory SIAM Rev. 19 46–89
Dennis J E Jr and Schnabel R B (1981) A new derivation of symmetric positivedefinite 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 861 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 862 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 8320 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
5 Parameters
Note: this routine uses
reverse communication. Its use involves an initial entry, intermediate exits and reentries, and a final exit, as indicated by the
parameter IREVCM. Between intermediate exits and reentries,
all parameters must remain unchanged except those specified by the value of IREVCM.
 1: IREVCM – INTEGERInput/Output
On initial entry: must be set to $0$.
On intermediate exit:
specifies what values the calling program must assign to parameters of E04UFF/E04UFA before reentering the routine.
 ${\mathbf{IREVCM}}=1$
 Set OBJF to the value of the objective function $F\left(x\right)$.
 ${\mathbf{IREVCM}}=2$
 Set
${\mathbf{OBJGRD}}\left(\mathit{j}\right)$ to the value $\frac{\partial F}{\partial {x}_{\mathit{j}}}$ if available, for $\mathit{j}=1,2,\dots ,n$.
 ${\mathbf{IREVCM}}=3$
 Set OBJF and ${\mathbf{OBJGRD}}\left(j\right)$ as for ${\mathbf{IREVCM}}=1$ and ${\mathbf{IREVCM}}=2$.
 ${\mathbf{IREVCM}}=4$
 Set ${\mathbf{C}}\left(i\right)$ to the value of the constraint function ${c}_{i}\left(x\right)$, for each $i$ such that ${\mathbf{NEEDC}}\left(i\right)>0$.
 ${\mathbf{IREVCM}}=5$
 Set ${\mathbf{CJAC}}\left(i,j\right)$ to the value $\frac{\partial {c}_{i}}{\partial {x}_{j}}$ if available, for each $i$ such that ${\mathbf{NEEDC}}\left(i\right)>0$ and $j=1,2,\dots ,n$.
 ${\mathbf{IREVCM}}=6$
 Set ${\mathbf{C}}\left(i\right)$ and ${\mathbf{CJAC}}\left(i,j\right)$ as for ${\mathbf{IREVCM}}=4$ and ${\mathbf{IREVCM}}=5$.
On intermediate reentry:
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 E04UFF/E04UFA will take a final exit with
IFAIL set to this value of
IREVCM.
On final exit: ${\mathbf{IREVCM}}=0$.
Constraint:
${\mathbf{IREVCM}}\le 6$.
 2: N – INTEGERInput
On initial entry: $n$, the number of variables.
Constraint:
${\mathbf{N}}>0$.
 3: NCLIN – INTEGERInput
On initial entry: ${n}_{L}$, the number of general linear constraints.
Constraint:
${\mathbf{NCLIN}}\ge 0$.
 4: NCNLN – INTEGERInput
On initial entry: ${n}_{N}$, the number of nonlinear constraints.
Constraint:
${\mathbf{NCNLN}}\ge 0$.
 5: LDA – INTEGERInput
On initial entry: the first dimension of the array
A as declared in the (sub)program from which E04UFF/E04UFA is called.
Constraint:
${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NCLIN}}\right)$.
 6: LDCJ – INTEGERInput
On initial entry: the first dimension of the array
CJAC as declared in the (sub)program from which E04UFF/E04UFA is called.
Constraint:
${\mathbf{LDCJ}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NCNLN}}\right)$.
 7: LDR – INTEGERInput
On initial entry: the first dimension of the array
R as declared in the (sub)program from which E04UFF/E04UFA is called.
Constraint:
${\mathbf{LDR}}\ge {\mathbf{N}}$.
 8: A(LDA,$*$) – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array
A
must be at least
${\mathbf{N}}$ if
${\mathbf{NCLIN}}>0$, and at least
$1$ otherwise.
On initial entry: the
$i$th row of the matrix
${A}_{L}$ of general linear constraints in
(1) must be stored in
${\mathbf{A}}\left(\mathit{i},\mathit{j}\right)$, for
$\mathit{i}=1,2,\dots ,{\mathbf{NCLIN}}$ and
$\mathit{j}=1,2,\dots ,{\mathbf{N}}$.
That is, the
$\mathit{i}$th row contains the coefficients of the
$\mathit{i}$th general linear constraint, for
$\mathit{i}=1,2,\dots ,{\mathbf{NCLIN}}$.
If
${\mathbf{NCLIN}}=0$, the array
A is not referenced.
 9: BL(${\mathbf{N}}+{\mathbf{NCLIN}}+{\mathbf{NCNLN}}$) – REAL (KIND=nag_wp) arrayInput
 10: BU(${\mathbf{N}}+{\mathbf{NCLIN}}+{\mathbf{NCNLN}}$) – REAL (KIND=nag_wp) arrayInput
On initial entry:
BL must contain the lower bounds and
BU the upper bounds, for all the constraints in the following order. The first
$n$ elements of each array must contain the bounds on the variables, the next
${n}_{L}$ elements the bounds for the general linear constraints (if any) and the next
${n}_{N}$ elements the bounds for the general nonlinear constraints (if any). To specify a nonexistent lower bound (i.e.,
${l}_{j}=\infty $), set
${\mathbf{BL}}\left(j\right)\le \mathit{bigbnd}$, and to specify a nonexistent upper bound (i.e.,
${u}_{j}=+\infty $), set
${\mathbf{BU}}\left(j\right)\ge \mathit{bigbnd}$; the default value of
$\mathit{bigbnd}$ is
${10}^{20}$, but this may be changed by the optional parameter
Infinite Bound Size. To specify the
$j$th constraint as an
equality, set
${\mathbf{BL}}\left(j\right)={\mathbf{BU}}\left(j\right)=\beta $, say, where
$\left\beta \right<\mathit{bigbnd}$.
Constraints:
 ${\mathbf{BL}}\left(\mathit{j}\right)\le {\mathbf{BU}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{N}}+{\mathbf{NCLIN}}+{\mathbf{NCNLN}}$;
 if ${\mathbf{BL}}\left(j\right)={\mathbf{BU}}\left(j\right)=\beta $, $\left\beta \right<\mathit{bigbnd}$.
 11: ITER – INTEGERInput/Output
On intermediate reentry: must remain unchanged from a previous call to E04UFF/E04UFA.
On final exit: the number of major iterations performed.
 12: ISTATE(${\mathbf{N}}+{\mathbf{NCLIN}}+{\mathbf{NCNLN}}$) – INTEGER arrayInput/Output
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
$n$ elements of
ISTATE refer to the upper and lower bounds on the variables, the next
${n}_{L}$ elements refer to the upper and lower bounds on
${A}_{L}x$, and the next
${n}_{N}$ elements refer to the upper and lower bounds on
$c\left(x\right)$. Possible values for
${\mathbf{ISTATE}}\left(j\right)$ are as follows:
${\mathbf{ISTATE}}\left(j\right)$  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 ${\mathbf{BL}}\left(j\right)={\mathbf{BU}}\left(j\right)$. 
The values
$2$,
$1$ and
$4$ are also acceptable but will be modified by the routine. If E04UFF/E04UFA 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 routine also adjusts (if necessary) the values supplied in
X to be consistent with
ISTATE.
Constraint:
$2\le {\mathbf{ISTATE}}\left(\mathit{j}\right)\le 4$, for $\mathit{j}=1,2,\dots ,{\mathbf{N}}+{\mathbf{NCLIN}}+{\mathbf{NCNLN}}$.
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
${\mathbf{ISTATE}}\left(j\right)$ is as follows:
${\mathbf{ISTATE}}\left(j\right)$  Meaning 
$2$  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. 
$1$  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. 
$\phantom{}0$  The constraint is satisfied to within the feasibility tolerance, but is not in the QP working set. 
$\phantom{}1$  This inequality constraint is included in the QP working set at its lower bound. 
$\phantom{}2$  This inequality constraint is included in the QP working set at its upper bound. 
$\phantom{}3$  This constraint is included in the QP working set as an equality. This value of ISTATE can occur only when ${\mathbf{BL}}\left(j\right)={\mathbf{BU}}\left(j\right)$. 
 13: C($*$) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array
C
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NCNLN}}\right)$.
On initial entry: need not be set.
On intermediate reentry: if
${\mathbf{IREVCM}}=4$ or
$6$ and
${\mathbf{NEEDC}}\left(i\right)>0$,
${\mathbf{C}}\left(i\right)$ must contain the value of the
$i$th constraint at
$x$. The remaining elements of
C, corresponding to the nonpositive elements of
NEEDC, are ignored.
On final exit: if
${\mathbf{NCNLN}}>0$,
${\mathbf{C}}\left(\mathit{i}\right)$ contains the value of the
$\mathit{i}$th nonlinear constraint function
${c}_{\mathit{i}}$ at the final iterate, for
$\mathit{i}=1,2,\dots ,{\mathbf{NCNLN}}$.
If
${\mathbf{NCNLN}}=0$, the array
C is not referenced.
 14: CJAC(LDCJ,$*$) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array
CJAC
must be at least
${\mathbf{N}}$ if
${\mathbf{NCNLN}}>0$, and at least
$1$ otherwise.
On initial entry: in general,
CJAC need not be initialized before the call to E04UFF/E04UFA. However, if the optional parameter
${\mathbf{Derivative\; Level}}=2$ or
$3$, you may optionally set the constant elements of
CJAC. Such constant elements need not be reassigned on subsequent intermediate exits.
If all elements of the constraint Jacobian are known (i.e.,
${\mathbf{Derivative\; Level}}=2$ or
$3$), 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 E04UFF/E04UFA 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 reentry: if
${\mathbf{IREVCM}}=5$ or
$6$ and
${\mathbf{NEEDC}}\left(i\right)>0$, the
$i$th row of
CJAC must contain the available elements of the vector
$\nabla {c}_{i}$ given by
where
$\frac{\partial {c}_{i}}{\partial {x}_{j}}$ is the partial derivative of the
$i$th constraint with respect to the
$j$th variable, evaluated at the point
$x$. The remaining rows of
CJAC, corresponding to nonpositive elements of
NEEDC, are ignored. The
$\mathit{i}$th row of the Jacobian should be stored in elements
${\mathbf{CJAC}}\left(\mathit{i},\mathit{j}\right)$, for
$\mathit{i}=1,2,\dots ,{\mathbf{NCNLN}}$ and
$\mathit{j}=1,2,\dots ,{\mathbf{N}}$.
Note that constant nonzero elements do affect the values of the constraints. Thus, if ${\mathbf{CJAC}}\left(i,j\right)$ is set to a constant value, it need not be reset during subsequent intermediate exits, but the value ${\mathbf{CJAC}}\left(i,j\right)\times {\mathbf{X}}\left(j\right)$ must nonetheless be added to ${\mathbf{C}}\left(i\right)$. For example, if ${\mathbf{CJAC}}\left(1,1\right)=2$ and ${\mathbf{CJAC}}\left(1,2\right)=5$, then the term $2\times {\mathbf{X}}\left(1\right)5\times {\mathbf{X}}\left(2\right)$ must be included in the definition of ${\mathbf{C}}\left(1\right)$.
It must be emphasized that, if
${\mathbf{Derivative\; Level}}=0$ or
$1$, 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
$x$ 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.
On final exit: if
${\mathbf{NCNLN}}>0$,
CJAC contains the Jacobian matrix of the nonlinear constraint functions at the final iterate, i.e.,
${\mathbf{CJAC}}\left(\mathit{i},\mathit{j}\right)$ contains the partial derivative of the
$\mathit{i}$th constraint function with respect to the
$\mathit{j}$th variable, for
$\mathit{i}=1,2,\dots ,{\mathbf{NCNLN}}$ and
$\mathit{j}=1,2,\dots ,{\mathbf{N}}$.
If
${\mathbf{NCNLN}}=0$, the array
CJAC is not referenced.
 15: CLAMDA(${\mathbf{N}}+{\mathbf{NCLIN}}+{\mathbf{NCNLN}}$) – REAL (KIND=nag_wp) arrayInput/Output
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,
${\mathbf{CLAMDA}}\left(\mathit{j}\right)$ 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
$\mathit{j}={\mathbf{N}}+{\mathbf{NCLIN}}+1,\dots ,{\mathbf{N}}+{\mathbf{NCLIN}}+{\mathbf{NCNLN}}$. The remaining elements need not be set. Note that if the
$j$th constraint is defined as ‘inactive’ by the initial value of the
ISTATE array (i.e.
${\mathbf{ISTATE}}\left(j\right)=0$),
${\mathbf{CLAMDA}}\left(j\right)$ should be zero; if the
$j$th constraint is an inequality active at its lower bound (i.e.
${\mathbf{ISTATE}}\left(j\right)=1$),
${\mathbf{CLAMDA}}\left(j\right)$ should be nonnegative; if the
$j$th constraint is an inequality active at its upper bound (i.e.
${\mathbf{ISTATE}}\left(j\right)=2$),
${\mathbf{CLAMDA}}\left(j\right)$ should be nonpositive. If necessary, the routine will modify
CLAMDA to match these rules.
On final exit: the values of the QP multipliers from the last QP subproblem. ${\mathbf{CLAMDA}}\left(j\right)$ should be nonnegative if ${\mathbf{ISTATE}}\left(j\right)=1$ and nonpositive if ${\mathbf{ISTATE}}\left(j\right)=2$.
 16: OBJF – REAL (KIND=nag_wp)Input/Output
On initial entry: need not be set.
On intermediate reentry: if
${\mathbf{IREVCM}}=1$ or
$3$,
OBJF must be set to the value of the objective function at
$x$.
On final exit: the value of the objective function at the final iterate.
 17: OBJGRD(N) – REAL (KIND=nag_wp) arrayInput/Output
On initial entry: need not be set.
On intermediate reentry: if
${\mathbf{IREVCM}}=2$ or
$3$,
OBJGRD must contain the available elements of the gradient evaluated at
$x$.
See also the description of the optional parameter
Verify.
On final exit: the gradient of the objective function at the final iterate (or its finite difference approximation).
 18: R(LDR,N) – REAL (KIND=nag_wp) arrayInput/Output
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
$R$ 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.
On final exit: if
${\mathbf{Hessian}}=\mathrm{NO}$,
R contains the upper triangular Cholesky factor
$R$ of
${Q}^{\mathrm{T}}\stackrel{~}{H}Q$, an estimate of the transformed and reordered Hessian of the Lagrangian at
$x$ (see
(6) in
Section 10.1).
If
${\mathbf{Hessian}}=\mathrm{YES}$,
R contains the upper triangular Cholesky factor
$R$ of
$H$, the approximate (untransformed) Hessian of the Lagrangian, with the variables in the natural order.
 19: X(N) – REAL (KIND=nag_wp) arrayInput/Output
On initial entry: an initial estimate of the solution.
On intermediate exit:
the point $x$ at which the objective function, constraint functions or their derivatives are to be evaluated.
On final exit: the final estimate of the solution.
 20: NEEDC($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NCNLN}}\right)$) – INTEGER arrayOutput
On intermediate exit:
if
${\mathbf{IREVCM}}\ge 4$,
NEEDC specifies the indices of the elements of
C and/or
CJAC that must be assigned. If
${\mathbf{NEEDC}}\left(i\right)>0$, then the
$i$th element of
C and/or the available elements of the
$i$th row of
CJAC must be evaluated at
$x$.
 21: IWORK(LIWORK) – INTEGER arrayCommunication Array
 22: LIWORK – INTEGERInput
On initial entry: the dimension of the array
IWORK as declared in the (sub)program from which E04UFF/E04UFA is called.
Constraint:
${\mathbf{LIWORK}}\ge 3\times {\mathbf{N}}+{\mathbf{NCLIN}}+2\times {\mathbf{NCNLN}}$.
 23: WORK(LWORK) – REAL (KIND=nag_wp) arrayCommunication Array
 24: LWORK – INTEGERInput
On initial entry: the dimension of the array
WORK as declared in the (sub)program from which E04UFF/E04UFA is called.
Constraints:
 if ${\mathbf{NCNLN}}=0$ and ${\mathbf{NCLIN}}=0$, ${\mathbf{LWORK}}\ge 21\times {\mathbf{N}}+2$;
 if ${\mathbf{NCNLN}}=0$ and ${\mathbf{NCLIN}}>0$, ${\mathbf{LWORK}}\ge 2\times {{\mathbf{N}}}^{2}+21\times {\mathbf{N}}+11\times {\mathbf{NCLIN}}+2$;
 if ${\mathbf{NCNLN}}>0$ and ${\mathbf{NCLIN}}\ge 0$, ${\mathbf{LWORK}}\ge 2\times {{\mathbf{N}}}^{2}+{\mathbf{N}}\times {\mathbf{NCLIN}}+2\times {\mathbf{N}}\times {\mathbf{NCNLN}}+21\times {\mathbf{N}}+11\times {\mathbf{NCLIN}}+\text{}22\times {\mathbf{NCNLN}}+1$.
The amounts of workspace provided and required may be (by default for E04UFF) output on the current advisory message unit (as defined by
X04ABF). 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
$1$. (E04UFF/E04UFA will then terminate with
${\mathbf{IFAIL}}={\mathbf{9}}$.)
 25: IFAIL – INTEGERInput/Output

Note: for E04UFA, IFAIL does not occur in this position in the parameter list. See the additional parameters described below.
On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if
${\mathbf{IFAIL}}\ne {\mathbf{0}}$ on exit, the recommended value is
$1$.
When the value $\mathbf{1}\text{ or}1$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
E04UFF/E04UFA returns with
${\mathbf{IFAIL}}={\mathbf{0}}$ if the iterates have converged to a point
$x$ that satisfies the firstorder Kuhn–Tucker conditions (see
Section 10.1) to the accuracy requested by the optional parameter
Optimality Tolerance. This has
$\text{default value}={\epsilon}_{r}^{0.8}$, where
${\epsilon}_{r}$ is the value of the optional parameter
Function Precision (
$\text{default value}={\epsilon}^{0.9}$, where
$\epsilon $ is the
machine precision). That is
${\mathbf{IFAIL}}={\mathbf{0}}$ when the projected gradient and active constraint residuals are negligible at
$x$.
You should check whether the following four conditions are satisfied:
(i) 
the final value of Norm Gz (see Section 8.1) is significantly less than that at the starting point; 
(ii) 
during the final major iterations, the values of Step and Mnr (see Section 8.1) are both one; 
(iii) 
the last few values of both Norm Gz and Violtn (see Section 8.1) become small at a fast linear rate; and 
(iv) 
Cond Hz (see Section 8.1) is small. 
If all these conditions hold,
$x$ is almost certainly a local minimum of
(1).
 Note: the following are additional parameters for specific use with E04UFA. Users of E04UFF therefore need not read the remainder of this description.
 25: CWSAV($5$) – CHARACTER(80) arrayCommunication Array
 26: LWSAV($120$) – LOGICAL arrayCommunication Array
 27: IWSAV($610$) – INTEGER arrayCommunication Array
 28: RWSAV($475$) – REAL (KIND=nag_wp) arrayCommunication Array
The arrays
LWSAV,
IWSAV,
RWSAV and
CWSAV must not be altered between calls to any of the routines
E04WBF, E04UFA,
E04UDA or
E04UEA.
 29: IFAIL – INTEGERInput/Output
Note: see the parameter description for
IFAIL above.
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Note: E04UFF/E04UFA may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}<0$
A negative value of
IFAIL indicates an exit from E04UFF/E04UFA because you set
${\mathbf{IREVCM}}<0$ during an intermediate exit. The value of
IFAIL will be the same as your setting of
IREVCM.
 ${\mathbf{IFAIL}}=1$
The final iterate
$x$ satisfies the firstorder Kuhn–Tucker conditions (see
Section 10.1) to the accuracy requested, but the sequence of iterates has not yet converged. E04UFF/E04UFA was terminated because no further improvement could be made in the merit function (see
Section 8.1).
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 5 for
${\mathbf{IFAIL}}={\mathbf{0}}$ are satisfied,
$x$ is likely to be a solution of
(1) even if
${\mathbf{IFAIL}}={\mathbf{1}}$.
 ${\mathbf{IFAIL}}=2$
E04UFF/E04UFA 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
$\sigma $, you should ensure that the value of the optional parameter
Linear Feasibility Tolerance is greater than
$\sigma $. For example, if all elements of
${A}_{L}$ are of order unity and are accurate to only three decimal places,
Linear Feasibility Tolerance should be at least
${10}^{3}$.
 ${\mathbf{IFAIL}}=3$
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 8.1). 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, E04UFF/E04UFA should be restarted at a different starting point.
 ${\mathbf{IFAIL}}=4$
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 E04UFF/E04UFA or, alternatively, rerun E04UFF/E04UFA 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 illconditioning as described under
${\mathbf{IFAIL}}={\mathbf{6}}$.
Note that illconditioning in the working set is sometimes resolved automatically by the algorithm, in which case performing additional iterations may be helpful. However, illconditioning in the Hessian approximation tends to persist once it has begun, so that allowing additional iterations without altering
${\mathbf{R}}$ is usually inadvisable. If the quasiNewton 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 8.1), it may be worthwhile to try a
Warm Start at the final point as suggested above.
 ${\mathbf{IFAIL}}=5$
Not used by this routine.
 ${\mathbf{IFAIL}}=6$
$x$ does not satisfy the firstorder Kuhn–Tucker conditions (see
Section 10.1), and no improved point for the merit function (see
Section 8.1) 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 (
$\text{default value}={\epsilon}_{r}^{0.8}$, where
${\epsilon}_{r}$ is the value of the optional parameter
Function Precision) is too small. In this case you should apply the four tests described under
${\mathbf{IFAIL}}={\mathbf{0}}$ 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 E04UFF/E04UFA 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
${\mathbf{IFAIL}}={\mathbf{7}}$ 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 8.1) are large.
Another possibility is that the search direction has become inaccurate because of illconditioning in the Hessian approximation or the matrix of constraints in the working set; either form of illconditioning tends to be reflected in large values of
Mnr (the number of iterations required to solve each QP subproblem; see
Section 8.1).
If the condition estimate of the projected Hessian (
Cond Hz; see
Section 8.1) is extremely large, it may be worthwhile rerunning E04UFF/E04UFA from the final point with the optional parameter
Warm Start. In this situation,
ISTATE and
CLAMDA should be left unaltered and
${\mathbf{R}}$ should be reset to the identity matrix.
If the matrix of constraints in the working set is illconditioned (i.e.,
Cond T is extremely large; see
Section 12), it may be helpful to run E04UFF/E04UFA 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
$T$, whose diagonals will be printed if
${\mathbf{Major\; Print\; Level}}\ge 30$.)
 ${\mathbf{IFAIL}}=7$
The usersupplied 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 firststep, 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 $x=0$ or $x=1$ 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 COMMON 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.
 ${\mathbf{IFAIL}}=8$
Not used by this routine.
 ${\mathbf{IFAIL}}=9$

An input parameter is invalid.
 $\mathbf{\text{Overflow}}$

If overflow occurs then either an element of $C$ is very large, or the singular values or singular vectors have been incorrectly supplied.
7 Accuracy
If
${\mathbf{IFAIL}}={\mathbf{0}}$ on final exit then the vector returned in the array
X is an estimate of the solution to an accuracy of approximately
Optimality Tolerance (
$\text{default value}={\epsilon}^{0.8}$, where
$\epsilon $ is the
machine precision).
8.1 Description of the Printed Output
This section describes the intermediate printout and final printout produced by E04UFF/E04UFA. The intermediate printout is a subset of the monitoring information produced by E04UFF/E04UFA at every iteration (see
Section 12). 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
${\mathbf{Major\; Print\; Level}}\ge 10$ (the default for E04UFF, by default no output is produced by E04UFA).
The following line of summary output (
$\text{}<80$ 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 $1$ in the later iterations, since theoretical analysis predicts that the correct active set will be identified near the solution
(see Section 10).
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 ${\alpha}_{k}$ taken along the computed search direction. On reasonably wellbehaved problems, the unit step (i.e., ${\alpha}_{k}=1$) 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 10.3).
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 E04UFF/E04UFA terminates with ${\mathbf{IFAIL}}={\mathbf{3}}$ (no feasible point could be found for the nonlinear constraints).
If there are no nonlinear constraints present (i.e., ${\mathbf{NCNLN}}=0$) then this entry contains Objective, the value of the objective function $F\left(x\right)$. The objective function will decrease monotonically to its optimal value when there are no nonlinear constraints.

Norm Gz 
is $\Vert {Z}^{\mathrm{T}}{g}_{\mathrm{FR}}\Vert $, the Euclidean norm of the projected gradient
(see Section 10.2).
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 ${H}_{Z}$ (${H}_{Z}={Z}^{\mathrm{T}}{H}_{\mathrm{FR}}Z={R}_{Z}^{\mathrm{T}}{R}_{Z}$; see (6)). The larger this number, the more difficult the problem.

M 
is printed if the quasiNewton update has been modified to ensure that the Hessian approximation is positive definite
(see Section 10.4).

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 $x$ is close to a Kuhn–Tucker point (see Section 10.1).

L 
is printed if the linesearch has produced a relative change in $x$ 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 $R$ indicates that the approximate Hessian is badly conditioned then the approximate Hessian is refactorized using column interchanges. If necessary, $R$ 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 $\mathit{j}$, for $\mathit{j}=1,2,\dots ,n$, 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 ${\mathbf{BL}}\left(j\right)\le \mathit{bigbnd}$.

Upper Bound 
is the upper bound specified for the variable. None indicates that ${\mathbf{BU}}\left(j\right)\ge \mathit{bigbnd}$.

Lagr Mult 
is the Lagrange multiplier for the associated bound. This will be zero if State is FR unless ${\mathbf{BL}}\left(j\right)\le \mathit{bigbnd}$ and ${\mathbf{BU}}\left(j\right)\ge \mathit{bigbnd}$, in which case the entry will be blank. If $x$ is optimal, the multiplier should be nonnegative if State is LL and nonpositive if State is UL.

Slack 
is the difference between the variable Value and the nearer of its (finite) bounds ${\mathbf{BL}}\left(j\right)$ and ${\mathbf{BU}}\left(j\right)$. A blank entry indicates that the associated variable is not bounded (i.e., ${\mathbf{BL}}\left(j\right)\le \mathit{bigbnd}$ and ${\mathbf{BU}}\left(j\right)\ge \mathit{bigbnd}$).

The meaning of the printout for linear and nonlinear constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’,
${\mathbf{BL}}\left(j\right)$ and
${\mathbf{BU}}\left(j\right)$ replaced by
${\mathbf{BL}}\left(n+j\right)$ and
${\mathbf{BU}}\left(n+j\right)$ respectively and with the following changes in the heading:
L Con 
gives the name (L) and index $\mathit{j}$, for $\mathit{j}=1,2,\dots ,{n}_{L}$, of the linear constraint.

N Con 
gives the name (N) and index ($\mathit{j}{n}_{L}$), for $\mathit{j}={n}_{L}+1,\dots ,{n}_{L}+{n}_{N}$, 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.
9 Example
This is based on Problem 71 in
Murtagh and Saunders (1983) and involves the minimization of the nonlinear function
subject to the bounds
to the general linear constraint
and to the nonlinear constraints
The initial point, which is infeasible, is
and
$F\left({x}_{0}\right)=16$.
The optimal solution (to five figures) is
and
$F\left({x}^{*}\right)=17.014$. One bound constraint and both nonlinear constraints are active at the solution.
9.1 Program Text
Note: the following programs illustrate the use of E04UFF and E04UFA.
Program Text (e04uffe.f90)
Program Text (e04ufae.f90)
9.2 Program Data
Program Data (e04uffe.d)
Program Data (e04ufae.d)
9.3 Program Results
Program Results (e04uffe.r)
Program Results (e04ufae.r)
Note: the remainder of this document is intended for more advanced users. Section 10 contains a detailed description of the algorithm which may be needed in order to understand Sections 11 and 12. Section 11 describes the optional parameters which may be set by calls to E04UDF/E04UDA and/or E04UEF/E04UEA. Section 12 describes the quantities which can be requested to monitor the course of the computation.
10 Algorithmic Details
This section contains a detailed description of the method used by E04UFF/E04UFA.
10.1 Overview
E04UFF/E04UFA is essentially identical to the subroutine 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
$C$ denote the
$m$ by
$n$ matrix of gradients of the active general linear and nonlinear constraints. The number of fixed variables will be denoted by
${n}_{\mathrm{FX}}$, with
${n}_{\mathrm{FR}}$ (
${n}_{\mathrm{FR}}=n{n}_{\mathrm{FX}}$) 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
$x$ is a
firstorder Kuhn–Tucker point for
(1) (see
Powell (1974)) if the following conditions hold:
(i) 
$x$ is feasible; 
(ii) 
there exist vectors $\xi $ and $\lambda $ (the Lagrange multiplier vectors for the bound and general constraints) such that
where $g$ is the gradient of $F$ evaluated at $x$ and ${\xi}_{j}=0$ if the $j$th variable is free. 
(iii) 
the Lagrange multiplier corresponding to an inequality constraint active at its lower bound must be nonnegative. It is nonpositive for an inequality constraint active at its upper bound. 
Let
$Z$ denote a matrix whose columns form a basis for the set of vectors orthogonal to the rows of
${C}_{\mathrm{FR}}$; i.e.,
${C}_{\mathrm{FR}}Z=0$. An equivalent statement of the condition
(2) in terms of
$Z$ is
The vector
${Z}^{\mathrm{T}}{g}_{\mathrm{FR}}$ is termed the
projected gradient of
$F$ at
$x$. Certain additional conditions must be satisfied in order for a firstorder Kuhn–Tucker point to be a solution of
(1) (see
Powell (1974)).
E04UFF/E04UFA 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 E04UFF/E04UFA involves
major and
minor iterations. The major iterations generate a sequence of iterates
$\left\{{x}_{k}\right\}$ that converge to
${x}^{*}$, a firstorder Kuhn–Tucker point of
(1). At a typical major iteration, the new iterate
$\stackrel{}{x}$ is defined by
where
$x$ is the current iterate, the nonnegative scalar
$\alpha $ is the
step length, and
$p$ 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
$p$ in
(3) is the solution of a quadratic programming subproblem of the form
where
$g$ is the gradient of
$F$ at
$x$, the matrix
$H$ is a positive definite quasiNewton approximation to the Hessian of the Lagrangian function (see
Section 10.4), and
${A}_{N}$ is the Jacobian matrix of
$c$ evaluated at
$x$. (Finite difference estimates may be used for
$g$ and
${A}_{N}$; see the optional parameter
Derivative Level.) Let
$l$ in
(1) be partitioned into three sections:
${l}_{B}$,
${l}_{L}$ and
${l}_{N}$, corresponding to the bound, linear and nonlinear constraints. The vector
$\stackrel{}{l}$ in
(4) is similarly partitioned and is defined as
where
$c$ is the vector of nonlinear constraints evaluated at
$x$. The vector
$\stackrel{}{u}$ 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 E04UFF/E04UFA; see
Section 12.) In E04UFF/E04UFA,
(4) is solved using
E04NCF/E04NCA. Since solving a quadratic program is itself an iterative procedure, the
minor iterations of E04UFF/E04UFA are the iterations of
E04NCF/E04NCA. (More details about solving the subproblem are given in
Section 10.2.)
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
$C$ denote the
$m$ by
$n$ matrix of gradients of the general linear and nonlinear constraints in the predicted active set. Firstly, we have available the
$TQ$ factorization of
${C}_{\mathrm{FR}}$:
where
$T$ is a nonsingular
$m$ by
$m$ reversetriangular matrix (i.e.,
${t}_{ij}=0$ if
$i+j<m$), and the nonsingular
${n}_{\mathrm{FR}}$ by
${n}_{\mathrm{FR}}$ matrix
${Q}_{\mathrm{FR}}$ is the product of orthogonal transformations (see
Gill et al. (1984b)). Secondly, we have the upper triangular Cholesky factor
$R$ of the
transformed and reordered Hessian matrix
where
$\stackrel{~}{H}$ is the Hessian
$H$ with rows and columns permuted so that the free variables are first and
$Q$ is the
$n$ by
$n$ matrix
with
${I}_{\mathrm{FX}}$ the identity matrix of order
${n}_{\mathrm{FX}}$. If the columns of
${Q}_{\mathrm{FR}}$ are partitioned so that
then the
${n}_{Z}$ (
${n}_{Z}\equiv {n}_{\mathrm{FR}}m$) columns of
$Z$ form a basis for the null space of
${C}_{\mathrm{FR}}$. The matrix
$Z$ is used to compute the projected gradient
${Z}^{\mathrm{T}}{g}_{\mathrm{FR}}$ at the current iterate. (The values
Nz and
Norm Gz printed by E04UFF/E04UFA give
${n}_{Z}$ and
$\Vert {Z}^{\mathrm{T}}{g}_{\mathrm{FR}}\Vert $, see
Section 12.)
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
${x}^{*}$. In E04UFF/E04UFA, 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 E04UFF/E04UFA also saves computation in factorizing
${C}_{\mathrm{FR}}$ and
${H}_{Q}$.
Once
$p$ has been computed, the major iteration proceeds by determining a step length
$\alpha $ that produces a ‘sufficient decrease’ in an augmented Lagrangian
merit function (see
Section 10.3). Finally, the approximation to the transformed Hessian matrix
${H}_{Q}$ is updated using a modified BFGS quasiNewton update (see
Section 10.4) to incorporate new curvature information obtained in the move from
$x$ to
$\stackrel{}{x}$.
On entry to E04UFF/E04UFA, an iterative procedure from
E04NCF/E04NCA is executed, starting with the usersupplied 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 E04UFF/E04UFA 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 usersupplied 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 E04UFF/E04UFA 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 quasiNewton update of the approximate Hessian of the Lagrangian function. 
These three procedures are described in more detail in
Sections 10.2 to
10.4.
10.2 Solution of the Quadratic Programming Subproblem
The search direction
$p$ is obtained by solving
(4) using
E04NCF/E04NCA (see
Gill et al. (1986)), which was specifically designed to be used within an SQP algorithm for nonlinear programming.
E04NCF/E04NCA is based on a twophase (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 subroutines. The twophase 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
$p$ denote the current estimate of the solution of
(4); the new iterate
$\stackrel{}{p}$ is defined by
where, as in
(3),
$\sigma $ is a nonnegative step length and
$d$ is a search direction.
At the beginning of each iteration of
E04NCF/E04NCA, a
working set is defined of constraints (general and bound) that are satisfied exactly. The vector
$d$ is then constructed so that the values of constraints in the working set remain
unaltered for any move along
$d$. For a bound constraint in the working set, this property is achieved by setting the corresponding element of
$d$ 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
$C$ denote the submatrix of rows of
corresponding to general constraints in the working set. The general constraints in the working set will remain unaltered if
which is equivalent to defining
${d}_{\mathrm{FR}}$ as
for some vector
${d}_{Z}$, where
$Z$ is the matrix associated with the
$TQ$ factorization
(5) of
${C}_{\mathrm{FR}}$.
The definition of
${d}_{Z}$ in
(10) depends on whether the current
$p$ is feasible. If not,
${d}_{Z}$ is zero except for an element
$\gamma $ in the
$j$th position, where
$j$ and
$\gamma $ are chosen so that the sum of infeasibilities is decreasing along
$d$. (For further details, see
Gill et al. (1986).) In the feasible case,
${d}_{Z}$ satisfies the equations
where
${R}_{Z}$ is the Cholesky factor of
${Z}^{\mathrm{T}}{H}_{\mathrm{FR}}Z$ and
$q$ is the gradient of the quadratic objective function
$\left(q=g+Hp\right)$. (The vector
${Z}^{\mathrm{T}}{q}_{\mathrm{FR}}$ is the projected gradient of the QP.) With
(11),
$p+d$ 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 nonvertices in the presence of constraint dependencies). During the optimality phase, a zero projected gradient implies that $p$ 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 $\delta $ of the order of the machine precision, the Lagrange multiplier ${\mu}_{j}$ corresponding to an inequality constraint in the working set is said to be optimal if ${\mu}_{j}\le \delta $ when the $j$th constraint is at its upper bound, or if ${\mu}_{j}\ge \delta $ 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 ${\mu}_{j}$ will satisfy $\left(1+\delta \right)\le {\mu}_{j}\le \delta $ for an inequality constraint at its upper bound, and $\delta \le {\mu}_{j}\le \left(1+\delta \right)$ for an inequality at its lower bound. The Lagrange multiplier for an equality constraint will satisfy $\left{\mu}_{j}\right\le 1+\delta $.
The choice of step length
$\sigma $ in the QP iteration
(8) is based on remaining feasible with respect to the satisfied constraints. During the optimality phase, if
$p+d$ is feasible,
$\sigma $ will be taken as unity. (In this case, the projected gradient at
$\stackrel{}{p}$ will be zero.) Otherwise,
$\sigma $ is set to
${\sigma}_{\mathrm{M}}$, 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 ${C}_{\mathrm{FR}}$: if the status of a general constraint changes, a row of ${C}_{\mathrm{FR}}$ is altered; if a bound constraint enters or leaves the working set, a column of ${C}_{\mathrm{FR}}$ changes. Explicit representations are recurred of the matrices $T$, ${Q}_{\mathrm{FR}}$ and $R$, and of the vectors ${Q}^{\mathrm{T}}q$ and ${Q}^{\mathrm{T}}g$.
10.3 The Merit Function
After computing the search direction as described in
Section 10.2, each major iteration proceeds by determining a step length
$\alpha $ in
(3) that produces a ‘sufficient decrease’ in the augmented Lagrangian merit function
where
$x$,
$\lambda $ and
$s$ vary during the linesearch. The summation terms in
(12) involve only the
nonlinear constraints. The vector
$\lambda $ is an estimate of the Lagrange multipliers for the nonlinear constraints of
(1). The nonnegative
slack variables
$\left\{{s}_{i}\right\}$ 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 nonnegative vector
$\rho $ 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
$\rho $ (the printed quantity
Penalty; see
Section 12) is generally nondecreasing, although each
${\rho}_{i}$ may be reduced a limited number of times.
The merit function
(12) and its global convergence properties are described in
Gill et al. (1986a).
10.4 The QuasiNewton Update
The matrix
$H$ in
(4) is a
positive definite quasiNewton approximation to the Hessian of the Lagrangian function. (For a review of quasiNewton methods, see
Dennis and Schnabel (1983).) At the end of each major iteration, a new Hessian approximation
$\stackrel{}{H}$ is defined as a ranktwo modification of
$H$. In E04UFF/E04UFA, the BFGS (Broyden–Fletcher–Goldfarb–Shanno) quasiNewton update is used:
where
$s=\stackrel{}{x}x$ (the change in
$x$).
In E04UFF/E04UFA,
$H$ is required to be positive definite. If
$H$ is positive definite,
$\stackrel{}{H}$ defined by
(13) will be positive definite if and only if
${y}^{\mathrm{T}}s$ is positive (see
Dennis and Moré (1977)). Ideally,
$y$ in
(13) would be taken as
${y}_{L}$, the change in gradient of the Lagrangian function
where
${\mu}_{N}$ denotes the QP multipliers associated with the nonlinear constraints of the original problem. If
${y}_{L}^{\mathrm{T}}s$ is not sufficiently positive, an attempt is made to perform the update with a vector
$y$ of the form
where
${\omega}_{i}\ge 0$. If no such vector can be found, the update is performed with a scaled
${y}_{L}$. In this case,
M is printed to indicate that the update was modified.
Rather than modifying
$H$ itself, the Cholesky factor of the
transformed Hessian
${H}_{Q}$ (6) is updated, where
$Q$ is the matrix from
(5) associated with the active set of the QP subproblem. The update
(13) is equivalent to the following update to
${H}_{Q}$:
where
${y}_{Q}={Q}^{\mathrm{T}}y$, and
${s}_{Q}={Q}^{\mathrm{T}}s$. This update may be expressed as a
rankone update to
$R$ (see
Dennis and Schnabel (1981)).
11 Optional Parameters
Several optional parameters in E04UFF/E04UFA define choices in the problem specification or the algorithm logic. In order to reduce the complexity of using E04UFF/E04UFA 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 11.1.
Optional parameters may be specified by calling one, or both, of
E04UDF/E04UDA and
E04UEF/E04UEA before a call to E04UFF/E04UFA.
E04UDF/E04UDA reads options from an external options file, with
Begin and
End as the first and last lines respectively and each intermediate line defining a single optional parameter. For example,
Begin * Example options file
Print level = 5
End
The call
CALL E04UDF (IOPTNS, INFORM)
can then be used to read the file on unit
IOPTNS.
${\mathbf{INFORM}}=0$ on successful exit.
E04UDF/E04UDA should be consulted for a full description of this method of supplying optional parameters.
E04UEF/E04UEA can be called to supply options directly, one call being necessary for each optional parameter. For example,
CALL E04UEF ('Print Level = 1')
E04UEF/E04UEA 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 E04UFF/E04UFA (unless they define invalid values) and so remain in effect for subsequent calls to E04UFF/E04UFA.
11.1 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:
 the keywords, where the minimum abbreviation of each keyword is underlined (if no characters of an optional qualifier are underlined, the qualifier may be omitted);
 a parameter value,
where the letters $a$, $i\text{ and}r$ denote options that take character, integer and real values respectively;
 the default value, where the symbol $\epsilon $ is a generic notation for machine precision (see X02AJF), and ${\epsilon}_{r}$ denotes the relative precision of the objective function Function Precision.
Keywords and character values are case and white space insensitive.
Central Difference Interval  $r$  Default values are computed 
If the algorithm switches to central differences because the forwarddifference approximation is not sufficiently accurate then the value of
$r$ is used as the difference interval for every element of
$x$. 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 8.1). 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 $0.0$ and $1.0$ is appropriate.
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 E04UFF/E04UFA 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 5.
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. E04UFF/E04UFA 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
$2$,
$1\text{ or}4$ will be reset to zero, as will any elements which are set to
$3$ 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 E04UFF/E04UFA is called repeatedly to solve related problems.
Crash Tolerance  $r$  Default $\text{}=0.01$ 
This value is used in conjunction with the optional parameter
Cold Start (the default value) when E04UFF/E04UFA selects an initial working set. If
$0\le r\le 1$, the initial working set will include (if possible) bounds or general inequality constraints that lie within
$r$ of their bounds. In particular, a constraint of the form
${a}_{j}^{\mathrm{T}}x\ge l$ will be included in the initial working set if
$\left{a}_{j}^{\mathrm{T}}xl\right\le r\left(1+\leftl\right\right)$. If
$r<0$ or
$r>1$, the default value is used.
This special keyword may be used to reset all optional parameters to their default values.
Derivative Level  $i$  Default $\text{}=3$ 
This parameter indicates which derivatives are provided during intermediate exits. The possible choices for
$i$ are the following.
$i$ 
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 $i=3$ should be used whenever possible, since E04UFF/E04UFA is more reliable (and will usually be more efficient) when all derivatives are exact.
If
$i=0\text{ or}2$, E04UFF/E04UFA will estimate the unspecified elements of the objective gradient, using finite differences. The computation of finite difference approximations usually increases the total runtime, 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
$i=0\text{ or}1$, E04UFF/E04UFA 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, E04UFF/E04UFA will make an intermediate exit to the calling program twice: once to estimate the missing element in column
$2$, and again to estimate the two missing elements in column
$3$. (Since columns
$1$ and
$4$ 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 $i<0$ or $i>3$, the default value is used.
Difference Interval  $r$  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
$j$th variable is approximated using the interval
${\delta}_{j}$, where
${\delta}_{j}=r\left(1+\left{\hat{x}}_{j}\right\right)$, with
$\hat{x}$ 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
$\mathit{O}\left(r\right)$. 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 E04UFF/E04UFA determine constant elements in the objective and constraint gradients.
If you supply a value for this optional parameter, a small value between $0.0$ and $1.0$ is appropriate.
Feasibility Tolerance  $r$  Default $\text{}=\sqrt{\epsilon}$ 
The scalar
$r$ 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
$r$. If
$r<\epsilon $ or
$r\ge 1$, the default value is used. Using this keyword sets both optional parameters
Linear Feasibility Tolerance and
Nonlinear Feasibility Tolerance to
$r$, if
$\epsilon \le r<1$. (Additional details are given under the descriptions of these optional parameters.)
Function Precision  $r$  Default $\text{}={\epsilon}^{0.9}$ 
This parameter defines ${\epsilon}_{r}$, which is intended to be a measure of the accuracy with which the problem functions $F\left(x\right)$ and $c\left(x\right)$ can be computed. If $r<\epsilon $ or $r\ge 1$, the default value is used.
The value of
${\epsilon}_{r}$ should reflect the relative precision of
$1+\leftF\left(x\right)\right$; i.e.,
${\epsilon}_{r}$ acts as a relative precision when
$\leftF\right$ is large and as an absolute precision when
$\leftF\right$ is small. For example, if
$F\left(x\right)$ is typically of order
$1000$ and the first six significant digits are known to be correct, an appropriate value for
${\epsilon}_{r}$ would be
${10}^{6}$. In contrast, if
$F\left(x\right)$ is typically of order
${10}^{4}$ and the first six significant digits are known to be correct, an appropriate value for
${\epsilon}_{r}$ would be
${10}^{10}$. The choice of
${\epsilon}_{r}$ 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
${\epsilon}_{r}$ should be large enough so that E04UFF/E04UFA will not attempt to distinguish between function values that differ by less than the error inherent in the calculation.
Hessian   Default $=\mathrm{NO}$ 
This option controls the contents of the upper triangular matrix
$R$ (see
Section 5). E04UFF/E04UFA works exclusively with the
transformed and reordered Hessian
${H}_{Q}$ (6), and hence extra computation is required to form the Hessian itself. If
${\mathbf{Hessian}}=\mathrm{NO}$,
R contains the Cholesky factor of the transformed and reordered Hessian. If
${\mathbf{Hessian}}=\mathrm{YES}$, the Cholesky factor of the approximate Hessian itself is formed and stored in
R. You should select
${\mathbf{Hessian}}=\mathrm{YES}$ if a
Warm Start will be used for the next call to E04UFF/E04UFA.
Infinite Bound Size  $r$  Default $\text{}={10}^{20}$ 
If $r>0$, $r$ defines the ‘infinite’ bound $\mathit{bigbnd}$ in the definition of the problem constraints. Any upper bound greater than or equal to $\mathit{bigbnd}$ will be regarded as $+\infty $ (and similarly any lower bound less than or equal to $\mathit{bigbnd}$ will be regarded as $\infty $). If $r<0$, the default value is used.
Infinite Step Size  $r$  Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\mathit{bigbnd},{10}^{20}\right)$ 
If $r>0$, $r$ specifies the magnitude of the change in variables that is treated as a step to an unbounded solution. If the change in $x$ during an iteration would exceed the value of $r$, the objective function is considered to be unbounded below in the feasible region. If $r\le 0$, the default value is used.
Line Search Tolerance  $r$  Default $\text{}=0.9$ 
The value $r$ ($0\le r<1$) controls the accuracy with which the step $\alpha $ taken during each iteration approximates a minimum of the merit function along the search direction (the smaller the value of $r$, the more accurate the linesearch). The default value $r=0.9$ 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 $r<0$ or $r\ge 1$, the default value is used.
Linear Feasibility Tolerance  ${r}_{1}$  Default $\text{}=\sqrt{\epsilon}$ 
Nonlinear Feasibility Tolerance  ${r}_{2}$  Default $\text{}={\epsilon}^{0.33}$ or $\sqrt{\epsilon}$ 
The default value of ${r}_{2}$ is ${\epsilon}^{0.33}$ if ${\mathbf{Derivative\; Level}}=0$ or $1$, and $\sqrt{\epsilon}$ otherwise.
The scalars ${r}_{1}$ and ${r}_{2}$ 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 ${r}_{1}$. Similarly a nonlinear constraint is considered satisfied if its violation does not exceed ${r}_{2}$. If ${r}_{\mathit{m}}<\epsilon $ or ${r}_{\mathit{m}}\ge 1$, the default value is used, for $\mathit{m}=1,2$.
On entry to E04UFF/E04UFA, 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 ${r}_{1}$. All subsequent iterates will satisfy the linear constraints to within the same tolerance (unless ${r}_{1}$ is comparable to the finite difference interval).
For nonlinear constraints, the feasibility tolerance
${r}_{2}$ 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 E04UFF/E04UFA (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 $6$ decimal digits, it would be appropriate to specify ${r}_{1}$ as ${10}^{6}$.
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  $i$  Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,3\left(n+{n}_{L}\right)+10{n}_{N}\right)$ 
The value of $i$ specifies the maximum number of major iterations allowed before termination. Setting $i=0$ and ${\mathbf{Major\; Print\; Level}}>0$ means that the workspace needed will be computed and printed, but no iterations will be performed. If $i<0$, the default value is used.
Major Print Level  $i$  Default for E04UFF
$\text{}=10$ 
Print Level   Default for E04UFA
$\text{}=0$ 
The value of
$i$ controls the amount of printout produced by the major iterations of E04UFF/E04UFA, as indicated below. A detailed description of the printed output is given in
Section 8.1 (summary output at each major iteration and the final solution) and
Section 12 (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
X04ABF):
$i$ 
Output 
$\phantom{\ge 0}0$ 
No output. 
$\phantom{\ge 0}1$ 
The final solution only. 
$\phantom{\ge 0}5$ 
One line of summary output ($\text{}<80$ characters; see Section 8.1) for each major iteration (no printout of the final solution). 
$\text{}\ge 10$ 
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:
$i$ 
Output 
$\text{}<5$ 
No output. 
$\text{}\ge 5$ 
One long line of output ($\text{}>80$ characters; see Section 12) for each major iteration (no printout of the final solution). 
$\text{}\ge 20$ 
At each major iteration, the objective function, the Euclidean norm of the nonlinear constraint violations, the values of the nonlinear constraints (the vector $c$), the values of the linear constraints (the vector ${A}_{L}x$) and the current values of the variables (the vector $x$). 
$\text{}\ge 30$ 
At each major iteration, the diagonal elements of the matrix $T$ associated with the $TQ$ factorization (5) (see Section 10.1) of the QP working set and the diagonal elements of $R$, the triangular factor of the transformed and reordered Hessian (6) (see Section 10.1). 
If
${\mathbf{Major\; Print\; Level}}\ge 5$ and the
unit number defined by the optional parameter
Monitoring File is the same as that defined by
X04ABF,
then the summary output for each major iteration is suppressed.
Minor Iteration Limit  $i$  Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,3\left(n+{n}_{L}+{n}_{N}\right)\right)$ 
The value of $i$ specifies the maximum number of iterations for finding a feasible point with respect to the bounds and linear constraints (if any). The value of $i$ also specifies the maximum number of minor iterations for the optimality phase of each QP subproblem. If $i\le 0$, the default value is used.
Minor Print Level  $i$  Default $\text{}=0$ 
The value of
$i$ controls the amount of printout produced by the minor iterations of E04UFF/E04UFA (i.e., the iterations of the quadratic programming algorithm), as indicated below. A detailed description of the printed output is given in
Section 8.2 in E04NCF/E04NCA (summary output at each minor iteration and the final QP solution) and
Section 12 in E04NCF/E04NCA) (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
X04ABF):
$i$ 
Output 
$\phantom{\ge 0}0$ 
No output. 
$\phantom{\ge 0}1$ 
The final QP solution only. 
$\phantom{\ge 0}5$ 
One line of summary output ($\text{}<80$ characters; see Section 8.2 in E04NCF/E04NCA)
for each minor iteration (no printout of the final QP solution). 
$\text{}\ge 10$ 
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:
$i$ 
Output 
$\text{}<5$ 
No output. 
$\text{}\ge 5$ 
One long line of output ($\text{}>80$ characters; see
Section 8.2 in E04NCF/E04NCA)
for each minor iteration (no printout of the final QP solution). 
$\text{}\ge 20$ 
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. 
$\text{}\ge 30$ 
At each minor iteration, the diagonal elements of the matrix $T$ associated with the $TQ$ factorization (5) (see Section 10.1) of the QP working set and the diagonal elements of the Cholesky factor $R$ of the transformed Hessian (6) (see Section 10.1). 
If
${\mathbf{Minor\; Print\; Level}}\ge 5$ and
the unit number defined by the optional parameter
Monitoring File is the same as that defined by
X04ABF
then the summary output for each minor iteration is suppressed.
Monitoring File  $i$  Default $\text{}=1$ 
If
$i\ge 0$ and
${\mathbf{Major\; Print\; Level}}\ge 5$ or
$i\ge 0$ and
${\mathbf{Minor\; Print\; Level}}\ge 5$, monitoring information produced by E04UFF/E04UFA at every iteration is sent to
a file with logical unit number $i$.
If
$i<0$ and/or
${\mathbf{Major\; Print\; Level}}<5$ and ${\mathbf{Minor\; Print\; Level}}<5$, no monitoring information is produced.
Optimality Tolerance  $r$  Default $\text{}={\epsilon}_{r}^{0.8}$ 
The parameter $r$ (${\epsilon}_{r}\le r<1$) specifies the accuracy to which you wish the final iterate to approximate a solution of the problem. Broadly speaking, $r$ indicates the number of correct figures desired in the objective function at the solution. For example, if $r$ is ${10}^{6}$ and E04UFF/E04UFA terminates successfully, the final value of $F$ should have approximately six correct figures. If $r<{\epsilon}_{r}$ or $r\ge 1$, the default value is used.
E04UFF/E04UFA will terminate successfully if the iterative sequence of
$x$ values is judged to have converged and the final point satisfies the firstorder Kuhn–Tucker conditions (see
Section 10.1). The sequence of iterates is considered to have converged at
$x$ if
where
$p$ is the search direction and
$\alpha $ the step length from
(3). An iterate is considered to satisfy the firstorder conditions for a minimum if
and
where
${Z}^{\mathrm{T}}{g}_{\mathrm{FR}}$ is the projected gradient (see
Section 10.1),
${g}_{\mathrm{FR}}$ is the gradient of
$F\left(x\right)$ with respect to the free variables,
${\mathit{res}}_{j}$ is the violation of the
$j$th active nonlinear constraint, and
$\mathit{ftol}$ is the
Nonlinear Feasibility Tolerance.
Start Objective Check At Variable  ${i}_{1}$  Default $\text{}=1$ 
Stop Objective Check At Variable  ${i}_{2}$  Default $\text{}=n$ 
Start Constraint Check At Variable  ${i}_{3}$  Default $\text{}=1$ 
Stop Constraint Check At Variable  ${i}_{4}$  Default $\text{}=n$ 
These keywords take effect only if ${\mathbf{Verify\; Level}}>0$. 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 $30$ elements of the objective gradient appeared to be correct in an earlier run, so that only element $31$ remains questionable, it is reasonable to specify ${\mathbf{Start\; Objective\; Check\; At\; Variable}}=31$. If the first $30$ variables appear linearly in the objective, so that the corresponding gradient elements are constant, the above choice would also be appropriate.
If ${i}_{2\mathit{m}1}\le 0$ or ${i}_{2\mathit{m}1}>\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,{i}_{2\mathit{m}}\right)$, the default value is used, for $\mathit{m}=1,2$. If ${i}_{2\mathit{m}}\le 0$ or ${i}_{2\mathit{m}}>n$, the default value is used, for $\mathit{m}=1,2$.
Step Limit  $r$  Default $\text{}=2.0$ 
If
$r>0,r$ specifies the maximum change in variables at the first step of the linesearch. In some cases, such as
$F\left(x\right)=a{e}^{bx}$ or
$F\left(x\right)=a{x}^{b}$, even a moderate change in the elements of
$x$ can lead to floating point overflow. The parameter
$r$ is therefore used to encourage evaluation of the problem functions at meaningful points. Given any major iterate
$x$, the first point
$\stackrel{~}{x}$ at which
$F$ and
$c$ are evaluated during the linesearch is restricted so that
The linesearch may go on and evaluate
$F$ and
$c$ at points further from
$x$ 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 8.1). If
L is printed for most of the iterations,
$r$ should be set to a larger value.
Wherever possible, upper and lower bounds on
$x$ should be used to prevent evaluation of nonlinear functions at wild values. The default value
${\mathbf{Step\; Limit}}=2.0$ should not affect progress on wellbehaved functions, but values such as
$0.1\text{ or}0.01$ 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
$r\le 0$, the default value is used.
Verify Level  $i$  Default $\text{}=0$ 
Verify Constraint 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
$i$ are as follows:
$i$ 
Meaning 
$1$ 
No checks are performed. 
$\phantom{}0$ 
Only a ‘cheap’ test will be performed. 
$\ge 1$ 
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
${\mathbf{Verify\; Level}}=0$ to
$3$ in several ways. For example, the objective gradient will be verified if
Verify,
${\mathbf{Verify}}=\mathrm{YES}$,
Verify Gradients,
Verify Objective Gradients or
${\mathbf{Verify\; Level}}=1$ is specified. The constraint gradients will be verified if
${\mathbf{Verify}}=\mathrm{YES}$ or
${\mathbf{Verify\; Level}}=2$ or
Verify is specified. Similarly, the objective and the constraint gradients will be verified if
${\mathbf{Verify}}=\mathrm{YES}$ or
${\mathbf{Verify\; Level}}=3$ or
Verify is specified.
If $0\le i\le 3$, gradients will be verified at the first point that satisfies the linear constraints and bounds.
If $i=0$, 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
$1\le i\le 3$, 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 E04UFF/E04UFA to indicate whether or not each element appears to be correct.
If $10\le i\le 13$, the action is the same as for $i<10$, except that it will take place at the userspecified initial value of $x$.
If $i<1$ or $4\le i\le 9$ or $i>13$, the default value is used.
We suggest that ${\mathbf{Verify\; Level}}=3$ be used whenever a new calling program is being developed.
12 Description of Monitoring Information
This section describes the long line of output (
$\text{}>80$ characters) which forms part of the monitoring information produced by E04UFF/E04UFA. (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
${\mathbf{Major\; Print\; Level}}\ge 5$ and
${\mathbf{Monitoring\; File}}\ge 0$, the following line of output is produced at every major iteration of E04UFF/E04UFA
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 $1$ in the later iterations, since theoretical analysis predicts that the correct active set will be identified near the solution
(see Section 10).
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 ${\alpha}_{k}$ taken along the computed search direction. On reasonably wellbehaved problems, the unit step (i.e., ${\alpha}_{k}=1$) 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 10.3).
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 E04UFF/E04UFA terminates with ${\mathbf{IFAIL}}={\mathbf{3}}$ (no feasible point could be found for the nonlinear constraints).
If there are no nonlinear constraints present (i.e., ${\mathbf{NCNLN}}=0$) then this entry contains Objective, the value of the objective function $F\left(x\right)$. The objective function will decrease monotonically to its optimal value when there are no nonlinear constraints.

Norm Gz 
is $\Vert {Z}^{\mathrm{T}}{g}_{\mathrm{FR}}\Vert $, the Euclidean norm of the projected gradient
(see Section 10.2).
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 $Z$ (see Section 10.2). The value of Nz is the number of variables minus the number of constraints in the predicted active set; i.e., $\mathtt{Nz}=n\left(\mathtt{Bnd}+\mathtt{Lin}+\mathtt{Nln}\right)$.

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 $H$.

Cond Hz 
is a lower bound on the condition number of the projected Hessian approximation ${H}_{Z}$ (${H}_{Z}={Z}^{\mathrm{T}}{H}_{\mathrm{FR}}Z={R}_{Z}^{\mathrm{T}}{R}_{Z}$; 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 threeletter 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 E04UFF/E04UFA terminates with ${\mathbf{IFAIL}}={\mathbf{0}}$, you should check the solution carefully.

M 
is printed if the quasiNewton update has been modified to ensure that the Hessian approximation is positive definite
(see Section 10.4).

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 $x$ is close to a Kuhn–Tucker point (see Section 10.1).

L 
is printed if the linesearch has produced a relative change in $x$ 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.


On 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 $R$ 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.
On exit: if ${\mathbf{Hessian}}=\mathrm{NO}$, R contains the upper triangular Cholesky factor $R$ of ${Q}^{\mathrm{T}}\stackrel{~}{H}Q$, an estimate of the transformed and reordered Hessian of the Lagrangian at $x$ (see (6) in Section 10.1). If ${\mathbf{Hessian}}=\mathrm{YES}$, R contains the upper triangular Cholesky factor $R$ of $H$, the approximate (untransformed) Hessian of the Lagrangian, with the variables in the natural order.
