NAG FL Interface e04nff
(qp_dense_solve_old) e04nfa (qp_dense_solve)
Note:this routine usesoptional parametersto 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 10 of this document. If, however, you wish to reset some or all of the settings please refer to Section 11 for a detailed description of the algorithm, to Section 12 for a detailed description of the specification of the optional parameters and to Section 13 for a detailed description of the monitoring information produced by the routine.
e04nff/e04nfa solves general quadratic programming problems. It is not intended for large sparse problems.
e04nfa is a version of e04nff that has additional arguments in order to make it safe for use in multithreaded applications (see Section 5). The initialization routine e04wbfmust have been called before calling e04nfa.
e04nff/e04nfa is designed to solve a class of quadratic programming problems that are assumed to be stated in the following general form:
$$\underset{x\in {R}^{n}}{\mathrm{minimize}}\phantom{\rule{0.25em}{0ex}}f\left(x\right)\text{\hspace{1em} subject to \hspace{1em}}l\le \left(\begin{array}{c}x\\ Ax\end{array}\right)\le u\text{,}$$
where $A$ is an ${m}_{L}\times n$ matrix and $f\left(x\right)$ may be specified in a variety of ways depending upon the particular problem to be solved. The available forms for $f\left(x\right)$ are listed in Table 1, in which the prefixes FP, LP and QP stand for ‘feasible point’, ‘linear programming’ and ‘quadratic programming’ respectively and $c$ is an $n$-element vector.
There is no restriction on $H$ or ${H}^{\mathrm{T}}H$ apart from symmetry. If the quadratic function is convex, a global minimum is found; otherwise, a local minimum is found. The default problem type is QP2 and other objective functions are selected by using the optional parameter Problem Type. For problems of type FP, the objective function is omitted and the routine attempts to find a feasible point for the set of constraints.
The constraints involving $A$ are called the general constraints. Note that upper and lower bounds are specified for all the variables and for all the general constraints. An equality constraint can be specified by setting ${l}_{i}={u}_{i}$. 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.)
The defining feature of a quadratic function $f\left(x\right)$ is that the second-derivative matrix ${\nabla}^{2}f\left(x\right)$ (the Hessian matrix) is constant. For QP1 and QP2 (the default), ${\nabla}^{2}f\left(x\right)=H$; for QP3 and QP4, ${\nabla}^{2}f\left(x\right)={H}^{\mathrm{T}}H$; and for the LP case, ${\nabla}^{2}f\left(x\right)=0$. If $H$ is positive semidefinite, it is usually more efficient to use e04ncf/e04nca. If $H$ is defined as the zero matrix, e04nff/e04nfa will still attempt to solve the resulting linear programming problem; however, this can be accomplished more efficiently by setting the optional parameter ${\mathbf{Problem\; Type}}=\mathrm{LP}$, or by using e04mff/e04mfa instead.
You must supply an initial estimate of the solution.
In the QP case, you may supply $H$ either explicitly as an $m\times n$ matrix, or implicitly in a subroutine that computes the product $Hx$ or ${H}^{\mathrm{T}}Hx$ for any given vector $x$.
In general, a successful run of e04nff/e04nfa will indicate one of three situations:
(i)a minimizer has been found;
(ii)the algorithm has terminated at a so-called dead-point; or
(iii)the problem has no bounded solution.
If a minimizer is found, and ${\nabla}^{2}f\left(x\right)$ is positive definite or positive semidefinite, e04nff/e04nfa will obtain a global minimizer; otherwise, the solution will be a local minimizer (which may or may not be a global minimizer). A dead-point is a point at which the necessary conditions for optimality are satisfied but the sufficient conditions are not. At such a point, a feasible direction of decrease may or may not exist, so that the point is not necessarily a local solution of the problem. Verification of optimality in such instances requires further information, and is in general an NP-hard problem (see Pardalos and Schnitger (1988)). Termination at a dead-point can occur only if ${\nabla}^{2}f\left(x\right)$ is not positive definite. If ${\nabla}^{2}f\left(x\right)$ is positive semidefinite, the dead-point will be a weak minimizer (i.e., with a unique optimal objective value, but an infinite set of optimal $x$).
The method used by e04nff/e04nfa (see Section 11) is most efficient when many constraints or bounds are active at the solution.
4References
Gill P E, Hammarling S, Murray W, Saunders M A and Wright M H (1986) Users' guide for LSSOL (Version 1.0) Report SOL 86-1 Department of Operations Research, Stanford University
Gill P E and Murray W (1978) Numerically stable methods for quadratic programming Math. Programming14 349–372
Gill P E, Murray W, Saunders M A and Wright M H (1984) Procedures for optimization problems with a mixture of bounds and general linear constraints ACM Trans. Math. Software10 282–298
Gill P E, Murray W, Saunders M A and Wright M H (1989) A practical anti-cycling procedure for linearly constrained optimization Math. Programming45 437–474
Gill P E, Murray W, Saunders M A and Wright M H (1991) Inertia-controlling methods for general quadratic programming SIAM Rev.33 1–36
Gill P E, Murray W and Wright M H (1981) Practical Optimization Academic Press
Pardalos P M and Schnitger G (1988) Checking local optimality in constrained quadratic programming is NP-hard Operations Research Letters7 33–35
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of variables.
Constraint:
${\mathbf{n}}>0$.
2: $\mathbf{nclin}$ – IntegerInput
On entry: ${m}_{L}$, the number of general linear constraints.
Constraint:
${\mathbf{nclin}}\ge 0$.
3: $\mathbf{a}({\mathbf{lda}},*)$ – 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 $1$ if ${\mathbf{nclin}}=0$.
On entry: the
$\mathit{i}$th row of a must contain the coefficients of the $\mathit{i}$th general linear constraint, for $\mathit{i}=1,2,\dots ,{m}_{L}$.
5: $\mathbf{bl}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$ – Real (Kind=nag_wp) arrayInput
6: $\mathbf{bu}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$ – Real (Kind=nag_wp) arrayInput
On 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, and the next ${m}_{L}$ elements the bounds for the general linear 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 $\mathit{j}=1,2,\dots ,{\mathbf{n}}+{\mathbf{nclin}}$;
if ${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)=\beta $, $\left|\beta \right|<\mathit{bigbnd}$.
7: $\mathbf{cvec}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array cvec
must be at least
${\mathbf{n}}$ if the problem is of type LP, QP2 (the default) or QP4, and at least $0$ otherwise.
On entry: the coefficients of the explicit linear term of the objective function when the problem is of type LP, QP2 (the default) and QP4.
If the problem is of type FP, QP1, or QP3, cvec is not referenced.
8: $\mathbf{h}({\mathbf{ldh}},*)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array h
must be at least
${\mathbf{n}}$ if it is to be used to store $H$ explicitly, and at least $1$ otherwise.
On entry: may be used to store the quadratic term $H$ of the QP objective function if desired. In some cases, you need not use h to store $H$ explicitly (see the specification of subroutine qphess). The elements of h are referenced only by subroutine qphess. The number of rows of $H$ is denoted by $m$, whose default value is $n$. (The optional parameter Hessian Rows may be used to specify a value of $m<n$.)
If the default version of qphess is used and the problem is of type QP1 or QP2 (the default), the first $m$ rows and columns of h must contain the leading $m\times m$ rows and columns of the symmetric Hessian matrix $H$. Only the diagonal and upper triangular elements of the leading $m$ rows and columns of h are referenced. The remaining elements need not be assigned.
If the default version of qphess is used and the problem is of type QP3 or QP4, the first $m$ rows of h must contain an $m\times n$ upper trapezoidal factor of the symmetric Hessian matrix ${H}^{\mathrm{T}}H$. The factor need not be of full rank, i.e., some of the diagonal elements may be zero. However, as a general rule, the larger the dimension of the leading nonsingular sub-matrix of h, the fewer iterations will be required. Elements outside the upper trapezoidal part of the first $m$ rows of h need not be assigned.
If a non-default version of qphess is supplied, then in some cases it may be desirable to use a one-dimensional array to transmit data to qphess. (This is illustrated in the example program in Section 10 in e04ngf/e04nga.) h is then declared as an ldh by $1$ array, where ${\mathbf{ldh}}\ge {\mathbf{n}}\times ({\mathbf{n}}+1)/2$.
In other situations, it may be desirable to compute $Hx$ or ${H}^{\mathrm{T}}Hx$ without accessing h – for example, if $H$ or ${H}^{\mathrm{T}}H$ is sparse or has special structure. The arguments h and ldh may then refer to any convenient array.
If the problem is of type FP or LP, h is not referenced.
9: $\mathbf{ldh}$ – IntegerInput
On entry: the first dimension of the array h as declared in the (sub)program from which e04nff/e04nfa is called.
Constraints:
if the problem is of type QP1, QP2 (the default), QP3 or QP4, ${\mathbf{ldh}}\ge {\mathbf{n}}$ or at least the value of the optional parameter Hessian Rows;
if the problem is of type FP or LP, ${\mathbf{ldh}}\ge 1$.
10: $\mathbf{qphess}$ – Subroutine, supplied by the NAG Library or the user.External Procedure
In general, you need not provide a version of qphess, because a ‘default’ subroutine with name e04nfu/e54nfu is included in the Library. However, the algorithm of e04nff/e04nfa requires only the product of $H$ or ${H}^{\mathrm{T}}H$ and a vector $x$; and in some cases you may obtain increased efficiency by providing a version of qphess that avoids the need to define the elements of the matrices $H$ or ${H}^{\mathrm{T}}H$ explicitly.
qphess is not referenced if the problem is of type FP or LP, in which case qphess may be the routine e04nfu/e54nfu.
On entry: this is the same argument as supplied to this routine. See the description for the top level argument n.
2: $\mathbf{jthcol}$ – IntegerInput
On entry: specifies whether or not the vector $x$ is a column of the identity matrix.
${\mathbf{jthcol}}=j>0$
The vector $x$ is the $j$th column of the identity matrix, and hence $Hx$ or ${H}^{\mathrm{T}}Hx$ is the $j$th column of $H$ or ${H}^{\mathrm{T}}H$, respectively. This may in some cases require very little computation and qphess may be coded to take advantage of this. However special code is not necessary because $x$ is always stored explicitly in the array x.
${\mathbf{jthcol}}=0$
$x$ has no special form.
3: $\mathbf{h}({\mathbf{ldh}},*)$ – Real (Kind=nag_wp) arrayInput
On entry: this is the same argument as supplied to this routine. See the description for the top level argument h.
4: $\mathbf{ldh}$ – IntegerInput
On entry: this is the same argument as supplied to this routine. See the description for the top level argument ldh.
5: $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the vector $x$.
6: $\mathbf{hx}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the product $Hx$ if the problem is of type QP1 or QP2 (the default), or the product ${H}^{\mathrm{T}}Hx$ if the problem is of type QP3 or QP4.
Note: the following are additional arguments for specific use with e04nfa. Users of e04nff therefore need not read the remainder of this description.
8: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
qphess is called with the arguments iuser and ruser as supplied to e04nff/e04nfa. You should use the arrays iuser and ruser to supply information to qphess.
iwsav contains information that is required by the default routine e54nfu.
qphess must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04nff/e04nfa is called. Arguments denoted as Input must not be changed by this procedure.
Note:qphess should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04nff/e04nfa. If your code inadvertently does return any NaNs or infinities, e04nff/e04nfa is likely to produce unexpected results.
On entry: need not be set if the (default) optional parameter Cold Start is used.
If the optional parameter Warm Start has been chosen, istate specifies the desired status of the constraints at the start of the feasibility phase. More precisely, the first $n$ elements of istate refer to the upper and lower bounds on the variables, and the next ${m}_{L}$ elements refer to the general linear constraints (if any). Possible values for ${\mathbf{istate}}\left(j\right)$ are as follows:
${\mathbf{istate}}\left(\mathit{j}\right)$
Meaning
0
The corresponding constraint should not be in the initial working set.
1
The constraint should be in the initial working set at its lower bound.
2
The constraint should be in the initial working set at its upper bound.
3
The constraint should be in the initial working set as an equality. This value must not be specified unless ${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)$.
The values $\mathrm{-2}$, $\mathrm{-1}$ and $4$ are also acceptable but will be reset to zero by the routine. If e04nff/e04nfa has been called previously with the same values of n and nclin, istate already contains satisfactory information. (See also the description of the optional parameter Warm Start.) The routine also adjusts (if necessary) the values supplied in x to be consistent with istate.
Constraint:
$\mathrm{-2}\le {\mathbf{istate}}\left(\mathit{j}\right)\le 4$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}+{\mathbf{nclin}}$.
On exit: the status of the constraints in the working set at the point returned in x. The significance of each possible value of ${\mathbf{istate}}\left(j\right)$ is as follows:
${\mathbf{istate}}\left(\mathit{j}\right)$
Meaning
$\mathrm{-2}$
The constraint violates its lower bound by more than the feasibility tolerance.
$\mathrm{-1}$
The constraint violates its upper bound by more than the feasibility tolerance.
$\phantom{-}0$
The constraint is satisfied to within the feasibility tolerance, but is not in the working set.
$\phantom{-}1$
This inequality constraint is included in the working set at its lower bound.
$\phantom{-}2$
This inequality constraint is included in the working set at its upper bound.
$\phantom{-}3$
This constraint is included in the working set as an equality. This value of istate can occur only when ${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)$.
$\phantom{-}4$
This corresponds to optimality being declared with ${\mathbf{x}}\left(j\right)$ being temporarily fixed at its current value. This value of istate can occur only when ${\mathbf{ifail}}={\mathbf{1}}$ on exit.
12: $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: an initial estimate of the solution.
On exit: the point at which e04nff/e04nfa terminated. If ${\mathbf{ifail}}={\mathbf{0}}$, ${\mathbf{1}}$ or ${\mathbf{4}}$, x contains an estimate of the solution.
13: $\mathbf{iter}$ – IntegerOutput
On exit: the total number of iterations performed.
14: $\mathbf{obj}$ – Real (Kind=nag_wp)Output
On exit: the value of the objective function at $x$ if $x$ is feasible, or the sum of infeasibilities at $x$ otherwise. If the problem is of type FP and $x$ is feasible, obj is set to zero.
15: $\mathbf{ax}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{nclin}})\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the final values of the linear constraints $Ax$.
16: $\mathbf{clamda}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the values of the Lagrange multipliers for each constraint with respect to the current working set. The first $n$ elements contain the multipliers for the bound constraints on the variables, and the next ${m}_{L}$ elements contain the multipliers for the general linear constraints (if any). If ${\mathbf{istate}}\left(j\right)=0$ (i.e., constraint $j$ is not in the working set), ${\mathbf{clamda}}\left(j\right)$ is zero. If $x$ is optimal, ${\mathbf{clamda}}\left(j\right)$ should be non-negative if ${\mathbf{istate}}\left(j\right)=1$, non-positive if ${\mathbf{istate}}\left(j\right)=2$ and zero if ${\mathbf{istate}}\left(j\right)=4$.
The amounts of workspace provided and required are (by default) output on the current advisory message unit (as defined by x04abf). As an alternative to computing liwork and lwork from the formulas given above, you may prefer to obtain appropriate values from the output of a preliminary run with liwork and lwork set to $1$. (e04nff/e04nfa will then terminate with ${\mathbf{ifail}}={\mathbf{6}}$.)
21: $\mathbf{ifail}$ – IntegerInput/Output
Note:for e04nfa, ifail does not occur in this position in the argument list. See the additional arguments described below.
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $\mathrm{-1}$ is recommended since useful values can be provided in some output arguments even when ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ 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).
e04nff/e04nfa returns with ${\mathbf{ifail}}={\mathbf{0}}$ if $x$ is a strong local minimizer, i.e., the reduced gradient (Norm Gz; see Section 9.2) is negligible, the Lagrange multipliers (Lagr Mult; see Section 9.2) are optimal and ${H}_{R}$ (the reduced Hessian of $f\left(x\right)$; see Section 11.2) is positive semidefinite.
Note: the following are additional arguments for specific use with e04nfa. Users of e04nff therefore need not read the remainder of this description.
25: $\mathbf{rwsav}\left(475\right)$ – Real (Kind=nag_wp) arrayCommunication Array
The arrays lwsav, iwsav and rwsavmust not be altered between calls to any of the routines e04nfa, e04ngaore04nha.
26: $\mathbf{ifail}$ – IntegerInput/Output
Note: see the argument description for ifail above.
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases e04nff/e04nfa may return useful information.
${\mathbf{ifail}}=1$
Iterations terminated at a dead point.
The necessary conditions for optimality are satisfied but the sufficient conditions are not. (The reduced gradient is negligible, the Lagrange multipliers are optimal, but ${H}_{R}$ is singular or there are some very small multipliers.) If ${\nabla}^{2}f\left(x\right)$ is not positive definite, $x$ is not necessarily a local solution of the problem and verification of optimality requires further information. If ${\nabla}^{2}f\left(x\right)$ is positive semidefinite or the problem is of type LP, $x$ gives the global minimum value of the objective function, but the final $x$ is not unique.
Optimal solution is not unique.
${\mathbf{ifail}}=2$
$\u27e8\mathit{\text{value}}\u27e9$ solution is unbounded.
The objective function is not bounded below in the feasible region. This value of ifail occurs if a step larger than Infinite Step Size ($\text{default value}={10}^{20}$) would have to be taken in order to continue the algorithm, or the next step would result in an element of $x$ having magnitude larger than optional parameter Infinite Bound Size ($\text{default value}={10}^{20}$).
${\mathbf{ifail}}=3$
Cannot satisfy the working set constraints to the accuracy requested.
No feasible point for the linear constraints.
It was not possible to satisfy all the constraints to within the feasibility tolerance. In this case, the constraint violations at the final $x$ will reveal a value of the tolerance for which a feasible point will exist – for example, when the feasibility tolerance for each violated constraint exceeds its Slack (see Section 9.2) at the final point. The modified problem (with an altered feasibility tolerance) may then be solved using a Warm Start. You should check that there are no constraint redundancies. If the data for the constraints are accurate only to the absolute precision $\sigma $, you should ensure that the value of the optional parameter Feasibility Tolerance ($\text{default value}=\sqrt{\epsilon}$, where $\epsilon $ is the machine precision) is greater than $\sigma $. For example, if all elements of $A$ are of order unity and are accurate only to three decimal places, the Feasibility Tolerance should be at least ${10}^{\mathrm{-3}}$.
${\mathbf{ifail}}=4$
Too many iterations.
The values of the optional parameters Feasibility Phase Iteration Limit ($\text{default value}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}(50,5(n+{m}_{L}))$) and Optimality Phase Iteration Limit ($\text{default value}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}(50,5(n+{m}_{L}))$) may be too small. If the method appears to be making progress (e.g., the objective function is being satisfactorily reduced), either increase the iterations limit and rerun e04nff/e04nfa or, alternatively, rerun e04nff/e04nfa using the Warm Start facility to specify the initial working set.
${\mathbf{ifail}}=5$
Reduced Hessian exceeds assigned dimension.
The algorithm needed to expand the reduced Hessian when it was already at its maximum dimension, as specified by the optional parameter Maximum Degrees of Freedom($\text{default value}=n$).
Rerun e04nff/e04nfa with a larger value (possibly using the Warm Start facility to specify the initial working set).
${\mathbf{ifail}}=6$
Not enough workspace to solve problem. Workspace provided is ${\mathbf{iwork}}\left(\u27e8\mathit{\text{value}}\u27e9\right)$ and ${\mathbf{work}}\left(\u27e8\mathit{\text{value}}\u27e9\right)$. To solve problem we need ${\mathbf{iwork}}\left(\u27e8\mathit{\text{value}}\u27e9\right)$ and ${\mathbf{work}}\left(\u27e8\mathit{\text{value}}\u27e9\right)$.
On entry, ${\mathbf{lda}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{nclin}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{nclin}})$.
On entry, ${\mathbf{ldh}}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$ and $m=\u27e8\mathit{\text{value}}\u27e9$ (where $m$ is the value of the optional parameter Hessian Rows).
Constraint: ${\mathbf{ldh}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}({\mathbf{n}},m)$.
On entry, ${\mathbf{ldh}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldh}}\ge 1$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}>0$.
On entry, ${\mathbf{nclin}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nclin}}\ge 0$.
On entry, the bounds on $\u27e8\mathit{\text{value}}\u27e9$ are inconsistent: ${\mathbf{bl}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{bu}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\u27e8\mathit{\text{value}}\u27e9$.
On entry, the bounds on linear constraint $\u27e8\mathit{\text{value}}\u27e9$ are inconsistent: ${\mathbf{bl}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{bu}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\u27e8\mathit{\text{value}}\u27e9$.
On entry, the bounds on nonlinear constraint $\u27e8\mathit{\text{value}}\u27e9$ are inconsistent: ${\mathbf{bl}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{bu}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\u27e8\mathit{\text{value}}\u27e9$.
On entry, the bounds on variable $\u27e8\mathit{\text{value}}\u27e9$ are inconsistent: ${\mathbf{bl}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{bu}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\u27e8\mathit{\text{value}}\u27e9$.
On entry, the equal bounds on $\u27e8\mathit{\text{value}}\u27e9$ are infinite, because ${\mathbf{bl}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\mathrm{beta}$ and ${\mathbf{bu}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\mathrm{beta}$, but $\left|\mathrm{beta}\right|\ge \mathrm{bigbnd}$: $\mathrm{beta}=\u27e8\mathit{\text{value}}\u27e9$ and $\mathrm{bigbnd}=\u27e8\mathit{\text{value}}\u27e9$.
On entry, the equal bounds on linear constraint $\u27e8\mathit{\text{value}}\u27e9$ are infinite, because ${\mathbf{bl}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\mathrm{beta}$ and ${\mathbf{bu}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\mathrm{beta}$, but $\left|\mathrm{beta}\right|\ge \mathrm{bigbnd}$: $\mathrm{beta}=\u27e8\mathit{\text{value}}\u27e9$ and $\mathrm{bigbnd}=\u27e8\mathit{\text{value}}\u27e9$.
On entry, the equal bounds on nonlinear constraint $\u27e8\mathit{\text{value}}\u27e9$ are infinite, because ${\mathbf{bl}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\mathrm{beta}$ and ${\mathbf{bu}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\mathrm{beta}$, but $\left|\mathrm{beta}\right|\ge \mathrm{bigbnd}$: $\mathrm{beta}=\u27e8\mathit{\text{value}}\u27e9$ and $\mathrm{bigbnd}=\u27e8\mathit{\text{value}}\u27e9$.
On entry, the equal bounds on variable $\u27e8\mathit{\text{value}}\u27e9$ are infinite, because ${\mathbf{bl}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\mathrm{beta}$ and ${\mathbf{bu}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\mathrm{beta}$, but $\left|\mathrm{beta}\right|\ge \mathrm{bigbnd}$: $\mathrm{beta}=\u27e8\mathit{\text{value}}\u27e9$ and $\mathrm{bigbnd}=\u27e8\mathit{\text{value}}\u27e9$.
On entry with a Cold Start, ${\mathbf{istate}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\u27e8\mathit{\text{value}}\u27e9$.
On entry with a Warm Start, ${\mathbf{istate}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\u27e8\mathit{\text{value}}\u27e9$.
If the printed output before the overflow error contains a warning about serious ill-conditioning in the working set when adding the $j$th constraint, it may be possible to avoid the difficulty by increasing the magnitude of the Feasibility Tolerance($\text{default value}=\sqrt{\epsilon}$, where $\epsilon $ is the machine precision) and rerunning the program. If the message recurs even after this change, the offending linearly dependent constraint (with index ‘$j$’) must be removed from the problem.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
e04nff/e04nfa implements a numerically stable active set strategy and returns solutions that are as accurate as the condition of the problem warrants on the machine.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
e04nff/e04nfa makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
This section contains some comments on scaling and a description of the printed output.
9.1Scaling
Sensible scaling of the problem is likely to reduce the number of iterations required and make the problem less sensitive to perturbations in the data, thus improving the condition of the problem. In the absence of better information it is usually sensible to make the Euclidean lengths of each constraint of comparable magnitude. See the E04 Chapter Introduction and Gill et al. (1981) for further information and advice.
9.2Description of the Printed Output
This section describes the intermediate printout and final printout produced by e04nff/e04nfa. The intermediate printout is a subset of the monitoring information produced by the routine at every iteration (see Section 13). You can control the level of printed output (see the description of the optional parameter Print Level).
Note that the intermediate printout and final printout are produced only if ${\mathbf{Print\; Level}}\ge 10$
(the default for e04nff, by default no output is produced by e04nfa).
The following line of summary output ($\text{}<80$ characters) is produced at every iteration. In all cases, the values of the quantities printed are those in effect on completion of the given iteration.
Itn
is the iteration count.
Step
is the step taken along the computed search direction. If a constraint is added during the current iteration, Step will be the step to the nearest constraint. When the problem is of type LP, the step can be greater than $1$ during the optimality phase.
Ninf
is the number of violated constraints (infeasibilities). This will be zero during the optimality phase.
Sinf/Objective
is the value of the current objective function. If $x$ is not feasible, Sinf gives a weighted sum of the magnitudes of constraint violations. If $x$ is feasible, Objective is the value of the objective function of (1). The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point.
During the optimality phase the value of the objective function will be nonincreasing. During the feasibility phase the number of constraint infeasibilities will not increase until either a feasible point is found or the optimality of the multipliers implies that no feasible point exists. Once optimal multipliers are obtained the number of infeasibilities can increase, but the sum of infeasibilities will either remain constant or be reduced until the minimum sum of infeasibilities is found.
Norm Gz
is $\Vert {Z}_{R}^{\mathrm{T}}{g}_{\mathrm{FR}}\Vert $, the Euclidean norm of the reduced gradient with respect to ${Z}_{R}$. During the optimality phase, this norm will be approximately zero after a unit step.
(See Sections 11.2 and 11.3.)
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 non-negative if State is LL and non-positive if State is UL.
Slack
is the difference between the variable Value and the nearer of its (finite) bounds ${\mathbf{bl}}\left(j\right)$ 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 general constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, ${\mathbf{bl}}\left(j\right)$ and ${\mathbf{bu}}\left(j\right)$ are replaced by ${\mathbf{bl}}\left(n+j\right)$ and ${\mathbf{bu}}\left(n+j\right)$ respectively, and with the following change 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.
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.
10Example
This example minimizes the quadratic function $f\left(x\right)={c}^{\mathrm{T}}x+\frac{1}{2}{x}^{\mathrm{T}}Hx$, where
Note: the remainder of this document is intended for more advanced users. Section 11 contains a detailed description of the algorithm which may be needed in order to understand Sections 12 and 13. Section 12 describes the optional parameters which may be set by calls to e04ngf/e04ngaand/ore04nhf/e04nha. Section 13 describes the quantities which can be requested to monitor the course of the computation.
11Algorithmic Details
This section contains a detailed description of the method used by e04nff/e04nfa.
11.1Overview
e04nff/e04nfa is based on an inertia-controlling method that maintains a Cholesky factorization of the reduced Hessian (see below). The method is based on that of Gill and Murray (1978), and is described in detail by Gill et al. (1991). Here we briefly summarise the main features of the method. Where possible, explicit reference is made to the names of variables that are arguments of e04nff/e04nfa or appear in the printed output. e04nff/e04nfa has two phases:
(i)finding an initial feasible point by minimizing the sum of infeasibilities (the feasibility phase), and
(ii)minimizing the quadratic objective function within the feasible region (the optimality phase).
The computations in both phases are performed by the same subroutines. The two-phase nature of the algorithm is reflected by changing the function being minimized from the sum of infeasibilities to the quadratic objective function. The feasibility phase does not perform the standard simplex method (i.e., it does not necessarily find a vertex), except in the LP case when ${m}_{L}\le n$. Once any iterate is feasible, all subsequent iterates remain feasible.
e04nff/e04nfa has been designed to be efficient when used to solve a sequence of related problems – for example, within a sequential quadratic programming method for nonlinearly constrained optimization (e.g., e04uff/e04ufaore04wdf). In particular, you may specify an initial working set (the indices of the constraints believed to be satisfied exactly at the solution); see the discussion of the optional parameter Warm Start.
In general, an iterative process is required to solve a quadratic program. (For simplicity, we shall always consider a typical iteration and avoid reference to the index of the iteration.) Each new iterate $\overline{x}$ is defined by
$$\overline{x}=x+\alpha p$$
(1)
where the step length$\alpha $ is a non-negative scalar and $p$ is called the search direction.
At each point $x$, a working set of constraints is defined to be a linearly independent subset of the constraints that are satisfied ‘exactly’ (to within the tolerance defined by the optional parameter Feasibility Tolerance). The working set is the current prediction of the constraints that hold with equality at the solution of a linearly constrained QP problem. The search direction is constructed so that the constraints in the working set remain unaltered for any value of the step length. For a bound constraint in the working set, this property is achieved by setting the corresponding element of the search direction to zero. Thus, the associated variable is fixed, and specification of the working set induces a partition of $x$ into fixed and free variables. During a given iteration, the fixed variables are effectively removed from the problem; since the relevant elements of the search direction are zero, the columns of $A$ corresponding to fixed variables may be ignored.
Let ${m}_{\mathrm{W}}$ denote the number of general constraints in the working set and let ${n}_{\mathrm{FX}}$ denote the number of variables fixed at one of their bounds (${m}_{\mathrm{W}}$ and ${n}_{\mathrm{FX}}$ are the quantities Lin and Bnd in the monitoring file output from e04nff/e04nfa; see Section 13). Similarly, let ${n}_{\mathrm{FR}}$ (${n}_{\mathrm{FR}}=n-{n}_{\mathrm{FX}}$) denote the number of free variables. At every iteration, the variables are reordered so that the last${n}_{\mathrm{FX}}$variables are fixed, with all other relevant vectors and matrices ordered accordingly.
11.2Definition of Search Direction
Let ${A}_{\mathrm{FR}}$ denote the ${m}_{\mathrm{W}}\times {n}_{\mathrm{FR}}$ sub-matrix of general constraints in the working set corresponding to the free variables and let ${p}_{\mathrm{FR}}$ denote the search direction with respect to the free variables only. The general constraints in the working set will be unaltered by any move along $p$ if
$${A}_{\mathrm{FR}}{p}_{\mathrm{FR}}=0\text{.}$$
(2)
In order to compute ${p}_{\mathrm{FR}}$, the $TQ$factorization of ${A}_{\mathrm{FR}}$ is used:
where $T$ is a nonsingular ${m}_{\mathrm{W}}\times {m}_{\mathrm{W}}$ upper triangular matrix (i.e., ${t}_{ij}=0$ if $i>j$), and the nonsingular ${n}_{\mathrm{FR}}\times {n}_{\mathrm{FR}}$ matrix ${Q}_{\mathrm{FR}}$ is the product of orthogonal transformations (see Gill et al. (1984)). If the columns of ${Q}_{\mathrm{FR}}$ are partitioned so that
where $Y$ is ${n}_{\mathrm{FR}}\times {m}_{\mathrm{W}}$, then the ${n}_{Z}$$({n}_{Z}={n}_{\mathrm{FR}}-{m}_{\mathrm{W}})$ columns of $Z$ form a basis for the null space of ${A}_{\mathrm{FR}}$. Let ${n}_{R}$ be an integer such that $0\le {n}_{R}\le {n}_{Z}$, and let ${Z}_{R}$ denote a matrix whose ${n}_{R}$ columns are a subset of the columns of $Z$. (The integer ${n}_{R}$ is the quantity Zr in the monitoring output from e04nff/e04nfa. In many cases, ${Z}_{R}$ will include all the columns of $Z$.) The direction ${p}_{\mathrm{FR}}$ will satisfy (2) if
where ${I}_{\mathrm{FX}}$ is the identity matrix of order ${n}_{\mathrm{FX}}$. Let ${H}_{Q}$ and ${g}_{Q}$ denote the $n\times n$transformed Hessian and transformed gradient
$${H}_{Q}={Q}^{\mathrm{T}}HQ\text{\hspace{1em} and \hspace{1em}}{g}_{Q}={Q}^{\mathrm{T}}(c+Hx)$$
and let the matrix of first ${n}_{R}$ rows and columns of ${H}_{Q}$ be denoted by ${H}_{R}$ and the vector of the first ${n}_{R}$ elements of ${g}_{Q}$ be denoted by ${g}_{R}$. The quantities ${H}_{R}$ and ${g}_{R}$ are known as the reduced Hessian and reduced gradient of $f\left(x\right)$, respectively. Roughly speaking, ${g}_{R}$ and ${H}_{R}$ describe the first and second derivatives of an unconstrained problem for the calculation of ${p}_{R}$.
At each iteration, a triangular factorization of ${H}_{R}$ is available. If ${H}_{R}$ is positive definite, ${H}_{R}={R}^{\mathrm{T}}R$, where $R$ is the upper triangular Cholesky factor of ${H}_{R}$. If ${H}_{R}$ is not positive definite, ${H}_{R}={R}^{\mathrm{T}}DR$, where $D=\mathrm{diag}(1,1,\dots ,1,\mu )$, with $\mu \le 0$.
The computation is arranged so that the reduced-gradient vector is a multiple of ${e}_{R}$, a vector of all zeros except in the last (i.e., ${n}_{R}$th) position. This allows the vector ${p}_{R}$ in (4) to be computed from a single back-substitution
$$R{p}_{R}=\gamma {e}_{R}$$
(5)
where $\gamma $ is a scalar that depends on whether or not the reduced Hessian is positive definite at $x$. In the positive definite case, $x+p$ is the minimizer of the objective function subject to the constraints (bounds and general) in the working set treated as equalities. If ${H}_{R}$ is not positive definite ${p}_{R}$ satisfies the conditions
$${p}_{R}^{\mathrm{T}}{H}_{R}{p}_{R}<0\text{\hspace{1em} and \hspace{1em}}{g}_{R}^{\mathrm{T}}{p}_{R}\le 0\text{,}$$
which allow the objective function to be reduced by any positive step of the form $x+\alpha p$.
11.3Main Iteration
If the reduced gradient is zero, $x$ is a constrained stationary point in the subspace defined by $Z$. During the feasibility phase, the reduced gradient will usually be zero only at a vertex (although it may be zero at non-vertices in the presence of constraint dependencies). During the optimality phase a zero reduced gradient implies that $x$ minimizes the quadratic objective when the constraints in the working set are treated as equalities. At a constrained stationary point, Lagrange multipliers ${\lambda}_{C}$ and ${\lambda}_{B}$ for the general and bound constraints are defined from the equations
$${A}_{\mathrm{FR}}^{\mathrm{T}}{\lambda}_{C}={g}_{\mathrm{FR}}\text{\hspace{1em} and \hspace{1em}}{\lambda}_{B}={g}_{\mathrm{FX}}-{A}_{\mathrm{FX}}^{\mathrm{T}}{\lambda}_{C}\text{.}$$
(6)
Given a positive constant $\delta $ of the order of the machine precision, a Lagrange multiplier ${\lambda}_{j}$ corresponding to an inequality constraint in the working set is said to be optimal if ${\lambda}_{j}\le \delta $ when the associated constraint is at its upper bound, or if ${\lambda}_{j}\ge -\delta $ when the associated constraint is at its lower bound. If a multiplier is nonoptimal, the objective function (either the true objective or the sum of infeasibilities) can be reduced by deleting the corresponding constraint (with index Jdel; see Section 13) from the working set.
If optimal multipliers occur during the feasibility phase and the sum of infeasibilities is nonzero, there is no feasible point, and you can force e04nff/e04nfa to continue until the minimum value of the sum of infeasibilities has been found; see the discussion of the optional parameter Minimum Sum of Infeasibilities. At such a point, the Lagrange multiplier ${\lambda}_{j}$ corresponding to an inequality constraint in the working set will be such that $-(1+\delta )\le {\lambda}_{j}\le \delta $ when the associated constraint is at its upper bound, and $-\delta \le {\lambda}_{j}\le (1+\delta )$ when the associated constraint is at its lower bound. Lagrange multipliers for equality constraints will satisfy $\left|{\lambda}_{j}\right|\le 1+\delta $.
If the reduced gradient is not zero, Lagrange multipliers need not be computed and the nonzero elements of the search direction $p$ are given by ${Z}_{R}{p}_{R}$ (see (4) and (5)). The choice of step length is influenced by the need to maintain feasibility with respect to the satisfied constraints. If ${H}_{R}$ is positive definite and $x+p$ is feasible, $\alpha $ will be taken as unity. In this case, the reduced gradient at $\overline{x}$ will be zero, and Lagrange multipliers are computed. Otherwise, $\alpha $ is set to ${\alpha}_{\mathrm{M}}$, the step to the ‘nearest’ constraint (with index Jadd; see Section 13), which is added to the working set at the next iteration.
Each change in the working set leads to a simple change to ${A}_{\mathrm{FR}}$: if the status of a general constraint changes, a row of ${A}_{\mathrm{FR}}$ is altered; if a bound constraint enters or leaves the working set, a column of ${A}_{\mathrm{FR}}$ changes. Explicit representations are recurred of the matrices $T$, ${Q}_{\mathrm{FR}}$ and $R$; and of vectors ${Q}^{\mathrm{T}}g$, and ${Q}^{\mathrm{T}}c$. The triangular factor $R$ associated with the reduced Hessian is only updated during the optimality phase.
One of the most important features of e04nff/e04nfa is its control of the conditioning of the working set, whose nearness to linear dependence is estimated by the ratio of the largest to smallest diagonal elements of the $TQ$ factor $T$ (the printed value Cond T; see Section 13). In constructing the initial working set, constraints are excluded that would result in a large value of Cond T.
e04nff/e04nfa includes a rigorous procedure that prevents the possibility of cycling at a point where the active constraints are nearly linearly dependent (see Gill et al. (1989)). The main feature of the anti-cycling procedure is that the feasibility tolerance is increased slightly at the start of every iteration. This not only allows a positive step to be taken at every iteration, but also provides, whenever possible, a choice of constraints to be added to the working set. Let ${\alpha}_{\mathrm{M}}$ denote the maximum step at which $x+{\alpha}_{\mathrm{M}}p$ does not violate any constraint by more than its feasibility tolerance. All constraints at a distance $\alpha $ ($\alpha \le {\alpha}_{\mathrm{M}}$) along $p$ from the current point are then viewed as acceptable candidates for inclusion in the working set. The constraint whose normal makes the largest angle with the search direction is added to the working set.
11.4Choosing the Initial Working Set
At the start of the optimality phase, a positive definite ${H}_{R}$ can be defined if enough constraints are included in the initial working set. (The matrix with no rows and columns is positive definite by definition, corresponding to the case when ${A}_{\mathrm{FR}}$ contains ${n}_{\mathrm{FR}}$ constraints.) The idea is to include as many general constraints as necessary to ensure that the reduced Hessian is positive definite.
Let ${H}_{Z}$ denote the matrix of the first ${n}_{Z}$ rows and columns of the matrix ${H}_{Q}={Q}^{\mathrm{T}}HQ$ at the beginning of the optimality phase. A partial Cholesky factorization is used to find an upper triangular matrix $R$ that is the factor of the largest positive definite leading sub-matrix of ${H}_{Z}$. The use of interchanges during the factorization of ${H}_{Z}$ tends to maximize the dimension of $R$. (The condition of $R$ may be controlled using the optional parameter Rank Tolerance.) Let ${Z}_{R}$ denote the columns of $Z$ corresponding to $R$, and let $Z$ be partitioned as $Z=\left(\begin{array}{cc}{Z}_{R}& {Z}_{A}\end{array}\right)$. A working set for which ${Z}_{R}$ defines the null space can be obtained by including the rows of${Z}_{A}^{\mathrm{T}}$ as ‘artificial constraints’. Minimization of the objective function then proceeds within the subspace defined by ${Z}_{R}$, as described in Section 11.2.
The artificially augmented working set is given by
so that ${p}_{\mathrm{FR}}$ will satisfy ${A}_{\mathrm{FR}}{p}_{\mathrm{FR}}=0$and${Z}_{A}^{\mathrm{T}}{p}_{\mathrm{FR}}=0$. By definition of the $TQ$ factorization,
${\overline{A}}_{\mathrm{FR}}$automatically satisfies the following:
and hence the $TQ$ factorization of (7) is available trivially from $T$ and ${Q}_{\mathrm{FR}}$ without additional expense.
The matrix ${Z}_{A}$ is not kept fixed, since its role is purely to define an appropriate null space; the $TQ$ factorization can, therefore, be updated in the normal fashion as the iterations proceed. No work is required to ‘delete’ the artificial constraints associated with ${Z}_{A}$ when ${Z}_{R}^{\mathrm{T}}{g}_{\mathrm{FR}}=0$, since this simply involves repartitioning ${Q}_{\mathrm{FR}}$. The ‘artificial’ multiplier vector associated with the rows of ${Z}_{A}^{\mathrm{T}}$ is equal to ${Z}_{A}^{\mathrm{T}}{g}_{\mathrm{FR}}$, and the multipliers corresponding to the rows of the ‘true’ working set are the multipliers that would be obtained if the artificial constraints were not present. If an artificial constraint is ‘deleted’ from the working set, an A appears alongside the entry in the Jdel column of the monitoring file output (see Section 13).
The number of columns in ${Z}_{A}$ and ${Z}_{R}$, the Euclidean norm of ${Z}_{R}^{\mathrm{T}}{g}_{\mathrm{FR}}$, and the condition estimator of $R$ appear in the monitoring file output as Art, Zr, Norm Gz and Cond Rz respectively (see Section 13).
Under some circumstances, a different type of artificial constraint isused when solving a linear program. Although the algorithm of e04nff/e04nfa does not usually perform simplex steps (in the traditional sense), there is one exception: a linear program with fewer general constraints than variables (i.e., ${m}_{L}\le n$). Use of the simplex method in this situation leads to savings in storage. At the starting point, the ‘natural’ working set (the set of constraints exactly or nearly satisfied at the starting point) is augmented with a suitable number of ‘temporary’ bounds, each of which has the effect of temporarily fixing a variable at its current value. In subsequent iterations, a temporary bound is treated as a standard constraint until it is deleted from the working set, in which case it is never added again. If a temporary bound is ‘deleted’ from the working set, an F (for ‘Fixed’) appears alongside the entry in the Jdel column of the monitoring file output (see Section 13).
12Optional Parameters
Several optional parameters in e04nff/e04nfa define choices in the problem specification or the algorithm logic. In order to reduce the number of formal arguments of e04nff/e04nfa 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 12.1.
Optional parameters may be specified by calling one, or both, of the routines e04ngf/e04ngaande04nhf/e04nha before a call to e04nff/e04nfa.
e04ngf/e04nga 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
Print Level = 5
End
The call
Call e04ngf/e04nga (ioptns, inform)
can then be used to read the file on unit ioptns.
inform will be zero on successful exit. e04ngf/e04nga should be consulted for a full description of this method of supplying optional parameters.
e04nhf/e04nha can be called to supply options directly, one call being necessary for each optional parameter.
For example,
Call e04nhf ('Print Level = 5')
e04nhf/e04nha should be consulted for a full description of this method of supplying optional parameters.
All optional parameters not specified by you are set to their default values. Optional parameters specified by you are unaltered by e04nff/e04nfa (unless they define invalid values) and so remain in effect for subsequent calls unless altered by you.
12.1Description 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 $a$, $i$ 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).
Keywords and character values are case and white space insensitive.
Check Frequency
$r$
Default $\text{}=50$
Every $i$th iteration, a numerical test is made to see if the current solution $x$ satisfies the constraints in the working set. If the largest residual of the constraints in the working set is judged to be too large, the current working set is refactorized and the variables are recomputed to satisfy the constraints more accurately. If $i\le 0$, the default value is used.
Cold Start
Default
Warm Start
This option specifies how the initial working set is chosen. With a Cold Start, e04nff/e04nfa chooses the initial working set based on the values of the variables and constraints at the initial point. Broadly speaking, the initial working set will include equality constraints and bounds or inequality constraints that violate or ‘nearly’ satisfy their bounds (to within Crash Tolerance).
With a Warm Start, you must provide a valid definition of every element of the array istate. e04nff/e04nfa 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 $\mathrm{-2}$, $\mathrm{-1}$ 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 e04nff/e04nfa 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 e04nff/e04nfa 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 $|{a}_{j}^{\mathrm{T}}x-l|\le r(1+\left|l\right|)$. If $r<0$ or $r>1$, the default value is used.
Defaults
This special keyword may be used to reset all optional parameters to their default values.
Expand Frequency
$i$
Default $\text{}=5$
This option is part of an anti-cycling procedure designed to guarantee progress even on highly degenerate problems.
The strategy is to force a positive step at every iteration, at the expense of violating the constraints by a small amount. Suppose that the value of the optional parameter Feasibility Tolerance is $\delta $. Over a period of $i$ iterations, the feasibility tolerance actually used by e04nff/e04nfa (i.e., the working feasibility tolerance) increases from $0.5\delta $ to $\delta $ (in steps of $0.5\delta /i$).
At certain stages the following ‘resetting procedure’ is used to remove constraint infeasibilities. First, all variables whose upper or lower bounds are in the working set are moved exactly onto their bounds. A count is kept of the number of nontrivial adjustments made. If the count is positive, iterative refinement is used to give variables that satisfy the working set to (essentially) machine precision. Finally, the working feasibility tolerance is reinitialized to $0.5\delta $.
If a problem requires more than $i$ iterations, the resetting procedure is invoked and a new cycle of $i$ iterations is started with $i$ incremented by $10$. (The decision to resume the feasibility phase or optimality phase is based on comparing any constraint infeasibilities with $\delta $.)
The resetting procedure is also invoked when e04nff/e04nfa reaches an apparently optimal, infeasible or unbounded solution, unless this situation has already occurred twice. If any nontrivial adjustments are made, iterations are continued.
If $i\le 0$, the default value is used. If $i\ge 9999999$, no anti-cycling procedure is invoked.
For problems of type FP, the scalar ${i}_{1}$ specifies the maximum number of iterations allowed before temination. Setting ${i}_{1}=0$ and ${\mathbf{Print\; Level}}>0$ means that the workspace needed will be computed and printed, but no iterations will be performed.
For problems of type LP, the maximum number of iterations allowed before temination is taken as $\mathrm{max}\phantom{\rule{0.125em}{0ex}}({i}_{1},{i}_{2})$. Setting ${i}_{1}=0$, ${i}_{2}=0$ and ${\mathbf{Print\; Level}}>0$ means that the workspace needed will be computed and printed, but no iterations will be performed.
For problems of type QP, the scalars ${i}_{1}$ and ${i}_{2}$ specify the maximum number of iterations allowed in the feasibility and optimality phases. Optimality Phase Iteration Limit is equivalent to Iteration Limit. Setting ${i}_{1}=0$ and ${\mathbf{Print\; Level}}>0$ means that the workspace needed will be computed and printed, but no iterations will be performed.
If ${i}_{1}<0$ or ${i}_{2}<0$, the default value is used.
Feasibility Tolerance
$r$
Default $\text{}=\sqrt{\epsilon}$
If $r\ge \epsilon $, $r$ defines the maximum acceptable absolute violation in each constraint at a ‘feasible’ point. For example, if the variables and the coefficients in the general constraints are of order unity, and the latter are correct to about $6$ decimal digits, it would be appropriate to specify $r$ as ${10}^{\mathrm{-6}}$. If $0\le r<\epsilon $, the default value is used.
e04nff/e04nfa attempts to find a feasible solution before optimizing the objective function. If the sum of infeasibilities cannot be reduced to zero, the optional parameter Minimum Sum of Infeasibilities can be used to find the minimum value of the sum. Let Sinf be the corresponding sum of infeasibilities. If Sinf is quite small, it may be appropriate to raise $r$ by a factor of $10$ or $100$. Otherwise, some error in the data should be suspected.
Note that a ‘feasible solution’ is a solution that satisfies the current constraints to within the tolerance $r$.
Hessian Rows
$i$
Default $\text{}=n$
Note that this option does not apply to problems of type FP or LP.
This specifies $m$, the number of rows of the Hessian matrix $H$. The default value of $m$ is $n$, the number of variables of the problem.
If the problem is of type QP, then $m$ will usually be $n$, the number of variables. However, a value of $m$ less than $n$ is appropriate for QP3 or QP4 if $H$ is an upper trapezoidal matrix with $m$ rows. Similarly, $m$ may be used to define the dimension of a leading block of nonzeros in the Hessian matrices of QP1 or QP2. In this case the last $n-m$ rows and columns of $H$ are assumed to be zero. In the QP case $m$ should not be greater than $n$; if it is, the last $m-n$ rows of $H$ are ignored.
If $i<0$ or $i>n$, the default value is used.
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.
If $r>0$, $r$ specifies the magnitude of the change in variables that will be considered a step to an unbounded solution. (Note that an unbounded solution can occur only when the Hessian is not positive definite.) If the change in $x$ during an iteration would exceed the value of $r$, then the objective function is considered to be unbounded below in the feasible region. If $r\le 0$, the default value is used.
Optional parameter List enables printing of each optional parameter specification as it is supplied. Nolist suppresses this printing.
Maximum Degrees of Freedom
$i$
Default $\text{}=n$
Note that this option does not apply to problems of type FP or LP.
This places a limit on the storage allocated for the triangular factor $R$ of the reduced Hessian ${H}_{R}$. Ideally, $i$ should be set slightly larger than the value of ${n}_{R}$ expected at the solution. It need not be larger than ${m}_{{\mathbf{n}}}+1$, where ${m}_{{\mathbf{n}}}$ is the number of variables that appear nonlinearly in the quadratic objective function. For many problems it can be much smaller than ${m}_{{\mathbf{n}}}$.
For quadratic problems, a minimizer may lie on any number of constraints, so that ${n}_{R}$ may vary between $1$ and $n$. The default value of $i$ is, therefore, the number of variables $n$. If Hessian Rows$m$ is specified, the default value of $i$ is the same number, $m$.
Minimum Sum of Infeasibilities
$a$
Default $=\mathrm{NO}$
If no feasible point exists for the constraints, then this option is used to control whether or not e04nff/e04nfa will calculate a point that minimizes the constraint violations. If ${\mathbf{Minimum\; Sum\; of\; Infeasibilities}}=\mathrm{NO}$, e04nff/e04nfa will terminate as soon as it is evident that no feasible point exists for the constraints. The final point will generally not be the point at which the sum of infeasibilities is minimized. If ${\mathbf{Minimum\; Sum\; of\; Infeasibilities}}=\mathrm{YES}$, e04nff/e04nfa will continue until the sum of infeasibilities is minimized.
Monitoring File
$i$
Default $\text{}=\mathrm{-1}$
If $i>6$ and ${\mathbf{Print\; Level}}\ge 5$, monitoring information produced by e04nff/e04nfa at every iteration is sent to a file with logical unit number $i$.
If $i<0$ and/or ${\mathbf{Print\; Level}}<5$, no monitoring information is produced.
Optimality Tolerance
$r$
Default $\text{}={\epsilon}^{0.5}$
If $r\ge \epsilon $, $r$ defines the tolerance used to determine if the bounds and general constraints have the right ‘sign’ for the solution to be judged to be optimal.
If $0\le r<\epsilon $, the default value is used.
Print Level
$i$
Default for e04nff$\text{}=10$
Default for e04nfa$\text{}=0$
The value of $i$ controls the amount of printout produced by e04nff/e04nfa, as indicated below. A detailed description of the printed output is given in Section 9.2 (summary output at each iteration and the final solution) and Section 13 (monitoring information at each iteration). If $i<0$, the default value is used.
The following printout is sent to the current advisory message unit (as defined by x04abf):
$\mathit{i}$
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 9.2) for each iteration (no printout of the final solution).
$\text{}\ge 10$
The final solution and one line of summary output for each iteration.
The following printout is sent to the unit number given by the optional parameter Monitoring File:
$\mathit{i}$
Output
$\text{}<5$
No output.
$\text{}\ge 5$
One long line of output ($\text{}>80$ characters; see Section 13) for each iteration (no printout of the final solution).
$\text{}\ge 20$
At each iteration: the Lagrange multipliers, the variables $x$, the constraint values $Ax$ and the constraint status (see istate).
$\text{}\ge 30$
At each iteration: the diagonal elements of the upper triangular matrix $T$ associated with the $TQ$ factorization (3) (see Section 11.2) of the working set and the diagonal elements of the upper triangular matrix $R$.
If ${\mathbf{Print\; Level}}\ge 5$ and the unit number defined by the optional parameter Monitoring File is the same as that defined by x04abf, the summary output is suppressed.
Problem Type
$a$
Default $=$ QP2
This option specifies the type of objective function to be minimized during the optimality phase. The following are the five optional keywords and the dimensions of the arrays that must be specified in order to define the objective function:
For problems of type FP the objective function is omitted and neither h nor cvec are referenced.
The following keywords are also acceptable. The minimum abbreviation of each keyword is underlined.
$\mathit{a}$
Option
Quadratic
QP2
Linear
LP
Feasible
FP
In addition, the keyword QP is equivalent to the default option QP2.
If $H=0$ (i.e., the objective function is purely linear), the efficiency of e04nff/e04nfa may be increased by specifying $a$ as LP.
Rank Tolerance
$r$
Default $\text{}=100\epsilon $
Note that this option does not apply to problems of type FP or LP.
This optional parameter enables you to control the condition number of the triangular factor $R$ (see Section 11). If ${\rho}_{i}$ denotes the function ${\rho}_{i}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\{\left|{R}_{11}\right|,\left|{R}_{22}\right|,\dots ,\left|{R}_{ii}\right|\}$, the dimension of $R$ is defined to be smallest index $i$ such that $\left|{R}_{i+1,i+1}\right|\le \sqrt{r}\left|{\rho}_{i+1}\right|$. If $r\le 0$, the default value is used.
13Description of Monitoring Information
This section describes the long line of output ($\text{}>80$ characters) which forms part of the monitoring information produced by e04nff/e04nfa. (See also the description of the optional parameters Monitoring File and Print Level.) You can control the level of printed output.
To aid interpretation of the printed results the following convention is used for numbering the constraints: indices $1$ through $n$ refer to the bounds on the variables and indices $n+1$ through $n+{m}_{L}$ refer to the general constraints. When the status of a constraint changes, the index of the constraint is printed, along with the designation L (lower bound), U (upper bound), E (equality), F (temporarily fixed variable) or A (artificial constraint).
When ${\mathbf{Print\; Level}}\ge 5$ and ${\mathbf{Monitoring\; File}}\ge 0$, the following line of output is produced at every iteration on the unit number specified by the Monitoring File. In all cases the values of the quantities printed are those in effect oncompletion of the given iteration.
Itn
is the iteration count.
Jdel
is the index of the constraint deleted from the working set. If Jdel is zero, no constraint was deleted.
Jadd
is the index of the constraint added to the working set. If Jadd is zero, no constraint was added.
Step
is the step taken along the computed search direction. If a constraint is added during the current iteration, Step will be the step to the nearest constraint. When the problem is of type LP, the step can be greater than $1$ during the optimality phase.
Ninf
is the number of violated constraints (infeasibilities). This will be zero during the optimality phase.
Sinf/Objective
is the value of the current objective function. If $x$ is not feasible, Sinf gives a weighted sum of the magnitudes of constraint violations. If $x$ is feasible, Objective is the value of the objective function of (1). The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point.
During the optimality phase the value of the objective function will be nonincreasing. During the feasibility phase the number of constraint infeasibilities will not increase until either a feasible point is found or the optimality of the multipliers implies that no feasible point exists. Once optimal multipliers are obtained the number of infeasibilities can increase, but the sum of infeasibilities will either remain constant or be reduced until the minimum sum of infeasibilities is found.
Bnd
is the number of simple bound constraints in the current working set.
Lin
is the number of general linear constraints in the current working set.
Art
is the number of artificial constraints in the working set, i.e., the number of columns of ${Z}_{A}$ (see Section 11.4).
Zr
is the number of columns of ${Z}_{1}$ (see Section 11.2). Zr is the dimension of the subspace in which the objective function is currently being minimized. The value of Zr is the number of variables minus the number of constraints in the working set; i.e., $\mathtt{Zr}=n-(\mathtt{Bnd}+\mathtt{Lin}+\mathtt{Art})$.
The value of ${n}_{Z}$, the number of columns of $Z$ (see Section 11.2) can be calculated as ${n}_{Z}=n-(\mathtt{Bnd}+\mathtt{Lin})$. A zero value of ${n}_{Z}$ implies that $x$ lies at a vertex of the feasible region.
Norm Gz
is $\Vert {Z}_{R}^{\mathrm{T}}{g}_{\mathrm{FR}}\Vert $, the Euclidean norm of the reduced gradient with respect to ${Z}_{R}$. During the optimality phase, this norm will be approximately zero after a unit step.
NOpt
is the number of nonoptimal Lagrange multipliers at the current point. NOpt is not printed if the current $x$ is infeasible or no multipliers have been calculated. At a minimizer, NOpt will be zero.
Min Lm
is the value of the Lagrange multiplier associated with the deleted constraint. If Min Lm is negative, a lower bound constraint has been deleted, if Min Lm is positive, an upper bound constraint has been deleted. If no multipliers are calculated during a given iteration Min Lm will be zero.
Cond T
is a lower bound on the condition number of the working set.
Cond Rz
is a lower bound on the condition number of the triangular factor $R$ (the Cholesky factor of the current reduced Hessian; see Section 11.2). If the problem is specified to be of type LP then Cond Rz is not printed.
Rzz
is the last diagonal element $\mu $ of the matrix $D$ associated with the ${R}^{\mathrm{T}}DR$ factorization of the reduced Hessian ${H}_{R}$ (see Section 11.2). Rzz is only printed if ${H}_{R}$ is not positive definite (in which case $\mu \ne 1$). If the printed value of Rzz is small in absolute value then ${H}_{R}$ is approximately singular. A negative value of Rzz implies that the objective function has negative curvature on the current working set.