For each option, we give a summary line, a description of the optional parameter and details of constraints.

The summary line contains:

Keywords and character values are case and white space insensitive.

When $\mathbf{Derivative\; Level}<3$, the central-difference interval $r$ is used near an optimal solution to obtain more accurate (but more expensive) estimates of gradients. Twice as many function evaluations are required compared to forward differencing. The interval used for the $j$th variable is $h_j=r\left(1+\left|x_j\right|\right)$. The resulting derivative estimates should be accurate to $\mathit{O}\left(r^2\right)$, unless the functions are badly scaled.

If you supply a value for this optional parameter, a small value between $0.0$ and $1.0$ is appropriate.

Every $i$th minor iteration after the most recent basis factorization, a numerical test is made to see if the current solution $x$ satisfies the general linear constraints (the linear constraints and the linearized nonlinear constraints, if any). The constraints are of the form $Ax-s=b$, where $s$ is the set of slack variables. To perform the numerical test, the residual vector $r=b-Ax+s$ is computed. If the largest component of $r$ is judged to be too large, the current basis is refactorized and the basic variables are recomputed to satisfy the general constraints more accurately. If $i\le 0$, the value of $i=99999999$ is used and effectively no checks are made.

$\mathbf{Check\; Frequency}=1$ is useful for debugging purposes, but otherwise this option should not be needed.

This option controls the specification of the initial working set in 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 e04wd 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 h as discussed in [Parameters]. istate values associated with bounds and linear constraints determine the initial working set of the procedure to find a feasible point with respect to the bounds and linear constraints. istate values associated with nonlinear constraints determine the initial working set of the first QP subproblem after such a feasible point has been found. e04wd 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 e04wd is called repeatedly to solve related problems.

If a Cold Start is specified, an internal Crash procedure is used to select an initial basis from certain rows and columns of the constraint matrix $\left(\begin{array}{cc}A& -I\end{array}\right)$. The optional parameter Crash Option $i$ determines which rows and columns of $A$ are eligible initially, and how many times the Crash procedure is called. Columns of $-I$ are used to pad the basis where necessary.

If $i\ge 1$, certain slacks on inequality rows are selected for the basis first. (If $i\ge 2$, numerical values are used to exclude slacks that are close to a bound). The Crash procedure then makes several passes through the columns of $A$, searching for a basis matrix that is essentially triangular. A column is assigned to ‘pivot’ on a particular row if the column contains a suitably large element in a row that has not yet been assigned. (The pivot elements ultimately form the diagonals of the triangular basis.) For remaining unassigned rows, slack variables are inserted to complete the basis.

The Crash Tolerance $r$ allows the starting Crash procedure to ignore certain ‘small’ nonzeros in each column of $A$. If $a_{\max }$ is the largest element in column $j$, other nonzeros of $a_{ij}$ in the columns are ignored if $\left|a_{ij}\right|\le a_{\max }\times r$. (To be meaningful, $r$ must be in the range $0\le r<1$.)

When $r>0.0$, the basis obtained by the Crash procedure may not be strictly triangular, but it is likely to be nonsingular and almost triangular. The intention is to obtain a starting basis containing more columns of $A$ and fewer (arbitrary) slacks. A feasible solution may be reached sooner on some problems.

For example, suppose the first $m$ columns of $A$ form the matrix shown under LU Factor Tolerance; i.e., a tridiagonal matrix with entries $-1$, $4$, $-1$. To help the Crash procedure choose all $m$ columns for the initial basis, we would specify a Crash Tolerance of $r$ for some value of $r>0.5$.

This special keyword may be used to reset all optional parameters to their default values.

Optional parameter Derivative Level specifies which nonlinear function gradients are known analytically and will be supplied to e04wd by user-supplied delegates objfun and confun.

The value $i=3$ should be used whenever possible. It is the most reliable and will usually be the most efficient.

If $i=0$ or $2$, e04wd will estimate the missing components of the objective gradient, using finite differences. This may simplify the coding of objfun. However, it could increase the total run-time substantially (since a special call to objfun is required for each missing element), and there is less assurance that an acceptable solution will be located. If the nonlinear variables are not well scaled, it may be necessary to specify a non-default optional parameter Difference Interval.

If $i=0$ or $1$, e04wd will estimate missing elements of the Jacobian. For each column of the Jacobian, one call to confun is needed to estimate all missing elements in that column, if any.

At times, central differences are used rather than forward differences. (This is not under your control.)

At each major iteration a linesearch is used to improve the merit function. Optional parameter Derivative Linesearch uses safeguarded cubic interpolation and requires both function and gradient values to compute estimates of the step ${\alpha }_k$. If some analytic derivatives are not provided, or optional parameter Nonderivative Linesearch is specified, e04wd employs a linesearch based upon safeguarded quadratic interpolation, which does not require gradient evaluations.

A nonderivative linesearch can be slightly less robust on difficult problems, and it is recommended that the default be used if the functions and derivatives can be computed at approximately the same cost. If the gradients are very expensive relative to the functions, a nonderivative linesearch may give a significant decrease in computation time.

If Nonderivative Linesearch is selected, e04wd signals the evaluation of the linesearch by calling objfun with $\mathbf{mode}=0$. If the potential saving provided by a nonderivative linesearch is to be realised, it is essential that objfun be coded so that derivatives are not computed when $\mathbf{mode}=0$.

This alters the interval $r$ used to estimate gradients by forward differences. It does so in the following circumstances:

–	in the interval (‘cheap’) phase of verifying the problem derivatives;
–	for verifying the problem derivatives;
–	for estimating missing derivatives.

In all cases, a derivative with respect to $x_j$ is estimated by perturbing that component of $x$ to the value $x_j+r\left(1+\left|x_j\right|\right)$, and then evaluating $F\left(x\right)$ or $c\left(x\right)$ at the perturbed point. The resulting gradient estimates should be accurate to $\mathit{O}\left(r\right)$ unless the functions are badly scaled. Judicious alteration of $r$ may sometimes lead to greater accuracy.

If you supply a value for this optional parameter, a small value between $0.0$ and $1.0$ is appropriate.

Optional parameters Dump File and Load File are similar to optional parameters Punch File and Insert File, but they record solution information in a manner that is more direct and more easily modified. A full description of information recorded in optional parameters Dump File and Load File is given in Gill et al. (2005a).

If $i_1>0$, the last solution obtained will be output to the file with unit number $i_1$.

If $i_2>0$, the Load File, containing basis information, will be read. The file will usually have been output previously as a Dump File. The file will not be accessed if optional parameters Old Basis File or Insert File are specified.

This keyword determines the initial weight $\gamma$ associated with the problem (11) (see [Treatment of Constraint Infeasibilities]).

At major iteration $k$, if elastic mode has not yet started, a scale factor ${\sigma }_k=1+{‖g\left(x_k\right)‖}_{\infty }$ is defined from the current objective gradient. Elastic mode is then started if the QP subproblem is infeasible, or the QP dual variables are larger in magnitude than ${\sigma }_{k}r$. The QP is resolved in elastic mode with $\gamma ={\sigma }_{k}r$.

Thereafter, major iterations continue in elastic mode until they converge to a point that is optimal for (11) (see [Treatment of Constraint Infeasibilities]). If the point is feasible for equation (1) $\left(v=w=0\right)$, it is declared locally optimal. Otherwise, $\gamma$ is increased by a factor of $10$ and major iterations continue. If $\gamma$ has already reached a maximum allowable value, equation (1) is declared locally infeasible.

This option is part of the anti-cycling procedure designed to make progress even on highly degenerate problems.

For linear models, the strategy is to force a positive step at every iteration, at the expense of violating the bounds on the variables by a small amount. Suppose that the optional parameter Minor Feasibility Tolerance is $\delta$. Over a period of $i$ iterations, the tolerance actually used by e04wd increases from $0.5\delta$ to $\delta$ (in steps of $0.5\delta /i$).

For nonlinear models, the same procedure is used for iterations in which there is only one superbasic variable. (Cycling can occur only when the current solution is at a vertex of the feasible region.) Thus, zero steps are allowed if there is more than one superbasic variable, but otherwise positive steps are enforced.

Increasing $i$ helps reduce the number of slightly infeasible nonbasic variables (most of which are eliminated during a resetting procedure). However, it also diminishes the freedom to choose a large pivot element (see optional parameter Pivot Tolerance).

At most $i$ basis changes will occur between factorizations of the basis matrix.

With linear programs, the basis factors are usually updated every iteration. The default $i$ is reasonable for typical problems. Higher values up to $i=100$ (say) may be more efficient on well-scaled problems.

When the objective function is nonlinear, fewer basis updates will occur as an optimum is approached. The number of iterations between basis factorizations will therefore increase. During these iterations a test is made regularly (according to the optional parameter Check Frequency) to ensure that the general constraints are satisfied. If necessary the basis will be refactorized before the limit of $i$ updates is reached.

The relative function precision ${\epsilon }_r$ is intended to be a measure of the relative accuracy with which the functions can be computed. For example, if $F\left(x\right)$ is computed as $1000.56789$ for some relevant $x$ and if the first $6$ significant digits are known to be correct, the appropriate value for ${\epsilon }_r$ would be $\text{1.0E−6}$.

(Ideally the functions $F\left(x\right)$ or $c_i\left(x\right)$ should have magnitude of order $1$. If all functions are substantially less than $1$ in magnitude, ${\epsilon }_r$ should be the absolute precision. For example, if $F\left(x\right)=\text{1.23456789E−4}$ at some point and if the first $6$ significant digits are known to be correct, the appropriate value for ${\epsilon }_r$ would be $\text{1.0E−10}$.)

The default value of ${\epsilon }_r$ is appropriate for simple analytic functions.

In some cases the function values will be the result of extensive computation, possibly involving a costly iterative procedure that can provide few digits of precision. Specifying an appropriate Function Precision may lead to savings, by allowing the linesearch procedure to terminate when the difference between function values along the search direction becomes as small as the absolute error in the values.

These options select the method for storing and updating the approximate Hessian. (e04wd uses a quasi-Newton approximation to the Hessian of the Lagrangian. A BFGS update is applied after each major iteration.)

If Hessian Full Memory is specified, the approximate Hessian is treated as a dense matrix and the BFGS updates are applied explicitly. This option is most efficient when the number of variables $n$ is not too large (say, less than $75$). In this case, the storage requirement is fixed and one can expect $1$-step Q-superlinear convergence to the solution.

Hessian Limited Memory should be used on problems where $n$ is very large. In this case a limited-memory procedure is used to update a diagonal Hessian approximation $H_r$ a limited number of times. (Updates are accumulated as a list of vector pairs. They are discarded at regular intervals after $H_r$ has been reset to their diagonal.)

If optional parameter Hessian Full Memory is in effect and $i$ BFGS updates have already been carried out, the Hessian approximation is reset to the identity matrix. (For certain problems, occasional resets may improve convergence, but in general they should not be necessary.)

Hessian Full Memory and $\mathbf{Hessian\; Frequency}=10$ have a similar effect to Hessian Limited Memory and $\mathbf{Hessian\; Updates}=10$ (except that the latter retains the current diagonal during resets).

If optional parameter Hessian Limited Memory is in effect and $i$ BFGS updates have already been carried out, all but the diagonal elements of the accumulated updates are discarded and the updating process starts again.

Broadly speaking, the more updates stored, the better the quality of the approximate Hessian. However, the more vectors stored, the greater the cost of each QP iteration. The default value is likely to give a robust algorithm without significant expense, but faster convergence can sometimes be obtained with significantly fewer updates (e.g., $i=5$).

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.

The value of $i$ specifies the maximum number of minor iterations allowed (i.e., iterations of the simplex method or the QP algorithm), summed over all major iterations. (See also the description of the optional parameter Minor Iterations Limit.)

This tolerance, $r$, controls the accuracy with which a step length will be located along the direction of search each iteration. At the start of each linesearch a target directional derivative for the merit function is identified. This parameter determines the accuracy to which this target value is approximated, and it must be a value in the range $0.0\le r\le 1.0$.

The default value $r=0.9$ requests just moderate accuracy in the linesearch.

If the nonlinear functions are cheap to evaluate, a more accurate search may be appropriate; try $r=0.1\text{, }0.01\text{ or }0.001$.

If the nonlinear functions are expensive to evaluate, a less accurate search may be appropriate. If all gradients are known, try $r=0.99$. (The number of major iterations might increase, but the total number of function evaluations may decrease enough to compensate.)

If not all gradients are known, a moderately accurate search remains appropriate. Each search will require only $1$–5 function values (typically), but many function calls will then be needed to estimate missing gradients for the next iteration.

For e04wd, normally each optional parameter specification is printed as it is supplied. Optional parameter Nolist may be used to suppress the printing and optional parameter List may be used to turn on printing.

The density tolerance, $r_1$, is used during $LU$ factorization of the basis matrix $B$. Columns of $L$ and rows of $U$ are formed one at a time, and the remaining rows and columns of the basis are altered appropriately. At any stage, if the density of the remaining matrix exceeds $r_1$, the Markowitz strategy for choosing pivots is terminated, and the remaining matrix is factored by a dense $LU$ procedure. Raising the density tolerance towards $1.0$ may give slightly sparser $LU$ factors, with a slight increase in factorization time.

The singularity tolerance, $r_2$, helps guard against ill-conditioned basis matrices. After $B$ is refactorized, the diagonal elements of $U$ are tested as follows: if $\left|u_{jj}\right|\le r_2$ or $\left|u_{jj}\right|<r_2{\displaystyle \underset{i}{\max }}\left|u_{ij}\right|$, the $j$th column of the basis is replaced by the corresponding slack variable. (This is most likely to occur after a restart.)

The values of $r_1$ and $r_2$ affect the stability of the basis factorization $B=LU$, during refactorization and updates respectively. The lower triangular matrix $L$ is a product of matrices of the form where the multipliers $\mu$ will satisfy $\left|\mu \right|\le r_i$. The default values of $r_1$ and $r_2$ usually strike a good compromise between stability and sparsity. They must satisfy $r_1$, $r_2\ge 1.0$.

For large and relatively dense problems, $r_1=10.0\text{ or }5.0$ (say) may give a useful improvement in stability without impairing sparsity to a serious degree.

For certain very regular structures (e.g., band matrices) it may be necessary to reduce $r_1\text{ and/or }{r}_2$ in order to achieve stability. For example, if the columns of $A$ include a sub-matrix of the form one should set both $r_1$ and $r_2$ to values in the range $1.0\le r_i<4.0$.

The $LU$ factorization implements a Markowitz-type search for pivots that locally minimize the fill-in subject to a threshold pivoting stability criterion. The default option is to use threshhold partial pivoting. The optional parameters LU Rook Pivoting and LU Complete Pivoting are more expensive than partial pivoting but are more stable and better at revealing rank, as long as LU Factor Tolerance is not too large (say $<2.0$). When numerical difficulties are encountered, e04wd automatically reduces the $LU$ tolerance towards $1.0$ and switches (if necessary) to rook or complete pivoting, before reverting to the default or specified options at the next refactorization (with System Information Yes, relevant messages are output to the Print File).

This tolerance, $r$, specifies how accurately the nonlinear constraints should be satisfied. The default value is appropriate when the linear and nonlinear constraints contain data to about that accuracy.

Let $v_{\max }$ be the maximum nonlinear constraint violation, normalized by the size of the solution, which is required to satisfy where $v_i$ is the violation of the $i$th nonlinear constraint $\left(i=1:n_L\right)$.

In the major iteration log (see [Minor Iteration Log], $v_{\max }$ appears as the quantity labelled ‘Feasible’. If some of the problem functions are known to be of low accuracy, a larger Major Feasibility Tolerance may be appropriate.

This tolerance, $r$, specifies the final accuracy of the dual variables. On successful termination, e04wd will have computed a solution $\left(x,s,\pi \right)$ such that where $c_j$ is an estimate of the complementarity slackness for variable $j$ where $j=1:n+n_L+n_N$. The values $c_i$ are computed from the final QP solution using the reduced gradients $d_j=g_j-{\pi }^{\mathrm{T}}{a}_j$ (where $g_j$ is the $j$th component of the objective gradient, $a_j$ is the associated column of the constraint matrix $\left(\begin{array}{cc}A& -I\end{array}\right)$, and $\pi$ is the set of QP dual variables):

In the Print File, $c_{\max }$ appears as the quantity labelled ‘Optimal’.

This is the maximum number of major iterations allowed. It is intended to guard against an excessive number of linearizations of the constraints. If $i=0$, optimality and feasibility are checked.

This controls the amount of output to the optional parameters Print File and Summary File at each major iteration. $\mathbf{Major\; Print\; Level}=0$ suppresses most output, except for error messages. $\mathbf{Major\; Print\; Level}=1$ gives normal output for linear and nonlinear problems, and $\mathbf{Major\; Print\; Level}=11$ gives additional details of the Jacobian factorization that commences each major iteration.

In general, the value being specified may be thought of as a binary number of the form where each letter stands for a digit that is either $0$ or $1$ as follows:

$s$	a single line that gives a summary of each major iteration. (This entry in $JFDXbs$ is not strictly binary since the summary line is printed whenever $JFDXbs\ge 1$);
$b$	basis statistics, i.e., information relating to the basis matrix whenever it is refactorized. (This output is always provided if $JFDXbs\ge 10$);
$X$	$x_k$, the nonlinear variables involved in the objective function or the constraints. These appear under the heading ‘Jacobian variables’;
$D$	${\pi }_k$, the dual variables for the nonlinear constraints. These appear under the heading ‘Multiplier estimates’;
$F$	$c\left(x_k\right)$, the values of the nonlinear constraint functions;
$J$	$J\left(x_k\right)$, the Jacobian matrix. This appears under the heading ‘$x$ and Jacobian’.

To obtain output of any items $JFDXbs$, set the corresponding digit to $1$, otherwise to $0$.

If $J=1$, the Jacobian matrix will be output column-wise at the start of each major iteration. Column $j$ will be preceded by the value of the corresponding variable $x_j$ and a key to indicate whether the variable is basic, superbasic or nonbasic. (Hence if $J=1$, there is no reason to specify $X=1$ unless the objective contains more nonlinear variables than the Jacobian.) A typical line of output is

 3 1.250000E+01 BS 1 1.00000E+00 4 2.00000E+00 

which would mean that $x_3$ is basic at value $12.5$, and the third column of the Jacobian has elements of $1.0$ and $2.0$ in rows $1$ and $4$.

This parameter limits the change in $x$ during a linesearch. It applies to all nonlinear problems, once a ‘feasible solution’ or ‘feasible subproblem’ has been found. A linesearch determines a step $\alpha$ over the range $0<\alpha \le \beta$, where $\beta$ is $1$ if there are nonlinear constraints, or is the step to the nearest upper or lower bound on $x$ if all the constraints are linear. Normally, the first step length tried is ${\alpha }_1=\min \left(1,\beta \right)$.

1.	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 components of $x$ can lead to floating-point overflow. The parameter $r$ is therefore used to define a limit $\stackrel{-}{\beta }=r\left(1+‖x‖\right)/‖p‖$ (where $p$ is the search direction), and the first evaluation of $f\left(x\right)$ is at the potentially smaller step length ${\alpha }_1=\min \left(1,\stackrel{-}{\beta },\beta \right)$.
2.	Wherever possible, upper and lower bounds on $x$ should be used to prevent evaluation of nonlinear functions at meaningless points. The optional parameter Major Step Limit provides an additional safeguard. The default value $r=2.0$ should not affect progress on well behaved problems, but setting $r=0.1\text{ or }0.01$ may be helpful when rapidly varying functions are present. A ‘good’ starting point may be required. An important application is to the class of nonlinear least squares problems.
3.	In cases where several local optima exist, specifying a small value for $r$ may help locate an optimum near the starting point.

The keywords Minimize and Maximize specify the required direction of optimization. It applies to both linear and nonlinear terms in the objective.

The keyword Feasible Point means ‘Ignore the objective function, while finding a feasible point for the linear and nonlinear constraints’. It can be used to check that the nonlinear constraints are feasible without altering the call to e04wd.

e04wd tries to ensure that all variables eventually satisfy their upper and lower bounds to within this tolerance, $r$. This includes slack variables. Hence, general linear constraints should also be satisfied to within $r$.

Feasibility with respect to nonlinear constraints is judged by the optional parameter Major Feasibility Tolerance (not by $r$).

If the bounds and linear constraints cannot be satisfied to within $r$, the problem is declared infeasible. If the corresponding sum of infeasibilities 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.

Nonlinear functions will be evaluated only at points that satisfy the bounds and linear constraints. If there are regions where a function is undefined, every attempt should be made to eliminate these regions from the problem.

For example, if $f\left(x\right)=\sqrt{x_1}+\log \left(x_2\right)$, it is essential to place lower bounds on both variables. If $r=\text{1.0E−6}$, the bounds $x_1\ge {10}^{-5}$ and $x_2\ge {10}^{-4}$ might be appropriate. (The log singularity is more serious. In general, keep $x$ as far away from singularities as possible.)

If $\mathbf{Scale\; Option}\ge 1$, feasibility is defined in terms of the scaled problem (since it is then more likely to be meaningful).

In reality, e04wd uses $r$ as a feasibility tolerance for satisfying the bounds on $x$ and $s$ in each QP subproblem. If the sum of infeasibilities cannot be reduced to zero, the QP subproblem is declared infeasible. e04wd is then in elastic mode thereafter (with only the linearized nonlinear constraints defined to be elastic). See the description of the optional parameter Elastic Weight.

If the number of minor iterations for the optimality phase of the QP subproblem exceeds $i$, then all nonbasic QP variables that have not yet moved are frozen at their current values and the reduced QP is solved to optimality.

Note that more than $i$ minor iterations may be necessary to solve the reduced QP to optimality. These extra iterations are necessary to ensure that the terminated point gives a suitable direction for the linesearch.

In the major iteration log (see [Minor Iteration Log]) a t at the end of a line indicates that the corresponding QP was artificially terminated using the limit $i$.

Compare with the optional parameter Iterations Limit, which defines an independent absolute limit on the total number of minor iterations (summed over all QP subproblems).

This controls the amount of output to the Print File and the Summary File during solution of the QP subproblems. The value of $i$ has the following effect:

New Basis File and Backup Basis File are sometimes referred to as basis maps. They contain the most compact representation of the state of each variable. They are intended for restarting the solution of a problem at a point that was reached by an earlier run. For nontrivial problems, it is advisable to save basis maps at the end of a run, in order to restart the run if necessary.

If $i_1>0$, a basis map will be saved on the file associated with unit $i_1$ every $i_3$th iteration. The first record of the file will contain the word PROCEEDING if the run is still in progress. A basis map will also be saved at the end of a run, with some other word indicating the final solution status.

Use of $i_2>0$ is intended as a safeguard against losing the results of a long run. Suppose that a New Basis File is being saved every $100$ (Save Frequency) iterations, and that e04wd is about to save such a basis at iteration $2000$. It is conceivable that the run may be interrupted during the next few milliseconds (in the middle of the save). In this case the Basis file will be corrupted and the run will have been essentially wasted.

To eliminate this risk, both a New Basis File and a Backup Basis File may be specified. The following would be suitable for the above example:

 Backup Basis File 11
 New Basis File 12 

The current basis will then be saved every $100$ iterations, first on the file associated with unit $12$ and then immediately on the file associated with unit $11$. If the run is interrupted at iteration $2000$ during the save on the file associated with unit $12$, there will still be a usable basis on the file associated with unit $11$ (corresponding to iteration $1900$).

Note that a new basis will be saved in New Basis File at the end of a run if it terminates normally, but it will not be saved in Backup Basis File. In the above example, if an optimum solution is found at iteration $2050$ (or if the iteration limit is $2050$), the final basis in the file associated with unit $12$ will correspond to iteration $2050$, but the last basis saved in the file associated with unit $11$ will be the one for iteration $2000$.

A full description of information recorded in New Basis File and Backup Basis File is given in Gill et al. (2005a).

This option causes early termination of the QP subproblems if the number of free variables has increased significantly since the first feasible point. If the number of new superbasics is greater than $i$, the nonbasic variables that have not yet moved are frozen and the resulting smaller QP is solved to optimality.

In the major iteration log (see [Major Iteration Log]), a t at the end of a line indicates that the QP was terminated early in this way.

If $i>0$, the basis maps information will be obtained from this file. The file will usually have been output previously as a New Basis File or Backup Basis File. A full description of information recorded in New Basis File and Backup Basis File is given in Gill et al. (2005a).

The file will not be acceptable if the number of rows or columns in the problem has been altered.

This parameter is recommended for large problems that have significantly more variables than constraints. It reduces the work required for each ‘pricing’ operation (where a nonbasic variable is selected to become superbasic). When $i=1$, all columns of the constraint matrix $\left(\begin{array}{cc}A& -I\end{array}\right)$ are searched. Otherwise, $A$ and $I$ are partitioned to give $i$ roughly equal segments $A_{\mathit{j}}$ and $I_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,i$. If the previous pricing search was successful on $A_{j-1}$ and $I_{j-1}$, the next search begins on the segments $A_j$ and $I_j$. (All subscripts here are modulo $i$.) If a reduced gradient is found that is larger than some dynamic tolerance, the variable with the largest such reduced gradient (of appropriate sign) is selected to become superbasic. If nothing is found, the search continues on the next segments $A_{j+1}$ and $I_{j+1}$, and so on.

For time-stage models having $t$ time periods, Partial Price $t$ (or $t/2$ or $t/3$) may be appropriate.

During the solution of QP subproblems, the pivot tolerance is used to prevent columns entering the basis if they would cause the basis to become almost singular.

When $x$ changes to $x+\alpha p$ for some search direction $p$, a ‘ratio test’ determines which component of $x$ reaches an upper or lower bound first. The corresponding element of $p$ is called the pivot element. Elements of $p$ are ignored (and therefore cannot be pivot elements) if they are smaller than the pivot tolerance $r$.

It is common for two or more variables to reach a bound at essentially the same time. In such cases, the Minor Feasibility Tolerance (say, $t$) provides some freedom to maximize the pivot element and thereby improve numerical stability. Excessively small values of $t$ should therefore not be specified. To a lesser extent, the Expand Frequency (say, $f$) also provides some freedom to maximize the pivot element. Excessively large values of $f$ should therefore not be specified.

If $i>0$, the following information is output to a file associated with unit $i$ during the solution of each problem:

–	a listing of the optional parameters;
–	some statistics about the problem;
–	the amount of storage available for the $LU$ factorization of the basis matrix;
–	notes about the initial basis resulting from a Crash procedure or a Basis file;
–	the iteration log;
–	basis factorization statistics;
–	the exit ifail condition and some statistics about the solution obtained;
–	the printed solution, if requested.

These items are described in [Further Comments] and [Description of Monitoring Information]. Further brief output may be directed to the Summary File.

If $i>0$, one line of the iteration log will be printed every $i$th iteration. A value such as $i=10$ is suggested for those interested only in the final solution. If $i\le 0$, the value of $i=99999999$ is used and effectively no checks are made.

$i=1\text{ or }2$ specifies minimization of ${‖x-x_0‖}_1$ or $\frac{1}{2}{‖x-x_0‖}_2^2$ when the starting point $x_0$ is changed to satisfy the linear constraints (where $x_0$ refers to nonlinear variables).

The Punch File from a previous run may be used as an Insert File for a later run on the same problem. A full description of information recorded in Insert File and Punch File is given in Gill et al. (2005a).

If $i_1>0$, the final solution obtained will be output to the file. For linear programs, this format is compatible with various commercial systems.

If $i_2>0$ the Insert File containing basis information will be read from unit $i_2$. The file will usually have been output previously as a Punch File. The file will not be accessed if Old Basis File is specified.

Specifies the active-set algorithm used to solve subproblem (11) (see [Treatment of Constraint Infeasibilities]). QPSolver Cholesky holds the full Cholesky factor $R$ of the reduced Hessian $Z^{\mathrm{T}}HZ$. As the QP iterations proceed, the dimension of $R$ changes with the number of superbasic variables. If the number of superbasic variables needs to increase beyond the value of Reduced Hessian Dimension, the reduced Hessian cannot be stored and the solver switches to QPSolver CG. The Cholesky solver is reactivated if the number of superbasics stabilizes at a value less than Reduced Hessian Dimension.

QPSolver QN solves the QP using a quasi-Newton method. In this case, $R$ is the factor of a quasi-Newton approximate Hessian.

QPSolver CG uses an active-set method similar to QPSolver QN, but uses the conjugate-gradient method to solve all systems involving the reduced Hessian.

The Cholesky QP solver is the most robust, but may require a significant amount of computation if there are many superbasics.

The quasi-Newton QP solver does not require computation of the exact $R$ at the start of the subproblem in (11). It may be appropriate when the number of superbasics is large but relatively few iterations are needed to reach a solution (e.g., if e04wd is called with a Warm Start).

The conjugate-gradient QP solver is appropriate for problems with many degrees of freedom (say, more than $2000$ superbasics).

This specifies that an $i$ by $i$ triangular matrix $R$ (to define the reduced Hessian according to $R^{\mathrm{T}}R=Z^{\mathrm{T}}HZ$) is to be available for use by the Cholesky QP solver.

Three scale options are available as follows:

Optional parameter Scale Tolerance affects how many passes might be needed through the constraint matrix. On each pass, the scaling procedure computes the ratio of the largest and smallest nonzero coefficients in each column: If ${\displaystyle \underset{j}{\max }}{\rho }_j$ is less than $r$ times its previous value, another scaling pass is performed to adjust the row and column scales. Raising $r$ from $0.9$ to $0.99$ (say) usually increases the number of scaling passes through $A$. At most $10$ passes are made. The value of $r$ should lie in the range $0<r<1$.

Scale Print causes the row scales $r\left(i\right)$ and column scales $c\left(j\right)$ to be printed to Print File, if System Information Yes has been specified. The scaled matrix coefficients are ${\stackrel{-}{a}}_{ij}=a_{ij}c\left(j\right)/r\left(i\right)$, and the scaled bounds on the variables and slacks are ${\stackrel{-}{l}}_j=l_j/c\left(j\right)$, ${\stackrel{-}{u}}_j=u_j/c\left(j\right)$, where $c\left(j\right)=r\left(j-n\right)$ if $j>n$.

If $i>0$, the final solution will be output to file $i$ (whether optimal or not). All numbers are printed in 1pe16.6 format.

To see more significant digits in the printed solution, it will sometimes be useful to make $i$ refer to Print File.

These keywords take effect only if $\mathbf{Verify\; Level}>0$. They may be used to contol the verification of gradient elements computed by function objfun and/or Jacobian elements computed by function confun. For eample, 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 $i_1>0$, a brief log will be output to the file associated with unit $i_1$, including one line of information every $i_2$th iteration. In an interactive environment, it is useful to direct this output to the terminal, to allow a run to be monitored online. (If something looks wrong, the run can be manually terminated.) Further details are given in [The Summary File].

This option places a limit on the storage allocated for superbasic variables. Ideally, $i$ should be set slightly larger than the ‘number of degrees of freedom’ expected at an optimal solution.

For nonlinear problems, the number of degrees of freedom is often called the ‘number of independent variables’. Normally, $i$ need not be greater than $n_N+1$, where $n_N$ is the number of nonlinear variables. For many problems, $i$ may be considerably smaller than $n_N$. This will save storage if $n_N$ is very large.

Normally e04wd prints the options file as it is being read, and then prints a complete list of the available keywords and their final values. The optional parameter Suppress Parameters tells e04wd not to print the full list.

This option prints additional information on the progress of major and minor iterations, and Crash statistics. See [Description of Monitoring Information].

If $i>0$, some timing information will be output to the Print file, if $\mathbf{Print\; File}>0$.

These parameters are intended to detect unboundedness in nonlinear problems. During a linesearch, $F$ is evaluated at points of the form $x+\alpha p$, where $x$ and $p$ are fixed and $\alpha$ varies. If $\left|F\right|$ exceeds $r_1$ or $\alpha$ exceeds $r_2$, iterations are terminated with the exit message $\mathbf{ifail}=5$.

If singularities are present, unboundedness in $F\left(x\right)$ may be manifested by a floating-point overflow (during the evaluation of $F\left(x+\alpha p\right)$), before the test against $r_1$ can be made.

Unboundedness in $x$ is best avoided by placing finite upper and lower bounds on the variables.

This option refers to finite difference checks on the derivatives computed by the user-supplied methods. Derivatives are checked at the first point that satisfies all bounds and linear constraints.

$\mathbf{Verify\; Level}=3$ should be specified whenever a new function method is being developed. Missing derivatives are not checked, so they result in no overhead.

This keyword defines an absolute limit on the magnitude of the maximum constraint violation, $r$, after the linesearch. On completion of the linesearch, the new iterate $x_{k+1}$ satisfies the condition where $x_0$ is the point at which the nonlinear constraints are first evaluated and $v_i\left(x\right)$ is the $i$th nonlinear constraint violation $v_i\left(x\right)=\max \left(0,l_i-c_i\left(x\right),c_i\left(x\right)-u_i\right)$.

The effect of this violation limit is to restrict the iterates to lie in an expanded feasible region whose size depends on the magnitude of $r$. This makes it possible to keep the iterates within a region where the objective is expected to be well defined and bounded below. If the obective is bounded below for all values of the variables, then $r$ may be any large positive value.