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.

Further details of other quantities not explicitly defined in this section may be found by consulting the document for e04uf.

If the algorithm switches to central differences because the forward-difference approximation is not sufficiently accurate, the value of $r$ is used as the difference interval for every element of $x$. The switch to central differences is indicated by C at the end of each line of intermediate printout produced by the major iterations (see [Description of the Printed Output]). The use of finite differences is discussed further under the optional parameter Difference Interval.

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

This option controls the specification of the initial working set in both the procedure for finding a feasible point for the linear constraints and bounds, and in the first QP subproblem thereafter. With a Cold Start, the first working set is chosen by e04us based on the values of the variables and constraints at the initial point. Broadly speaking, the initial working set will include equality constraints and bounds or inequality constraints that violate or ‘nearly’ satisfy their bounds (to within Crash Tolerance).

With a Warm Start, you must set the istate array and define clamda and r as discussed in [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. e04us 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 e04us is called repeatedly to solve related problems.

This value is used in conjunction with the optional parameter Cold Start (the default value) when e04us selects an initial working set. If $0\le r\le 1$, the initial working set will include (if possible) bounds or general inequality constraints that lie within $r$ of their bounds. In particular, a constraint of the form $a_j^{\mathrm{T}}x\ge l$ will be included in the initial working set if $\left|a_j^{\mathrm{T}}x-l\right|\le r\left(1+\left|l\right|\right)$. If $r<0$ or $r>1$, the default value is used.

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

This parameter indicates which derivatives are provided in user-supplied delegates objfun and confun. The possible choices for $i$ are the following.

The value $i=3$ should be used whenever possible, since e04us is more reliable (and will usually be more efficient) when all derivatives are exact.

If $i=0\text{ or }2$, e04us will approximate unspecified elements of the objective Jacobian, using finite differences. The computation of finite difference approximations usually increases the total run-time, since a call to objfun is required for each unspecified element. Furthermore, less accuracy can be attained in the solution (see Chapter 8 of Gill et al. (1981), for a discussion of limiting accuracy).

If $i=0\text{ or }1$, e04us will approximate unspecified elements of the constraint Jacobian. One call to confun is needed for each variable for which partial derivatives are not available. For example, if the constraint Jacobian has the form where ‘$*$’ indicates an element provided by you and ‘?’ indicates an unspecified element, e04us will call confun twice: once to estimate the missing element in column $2$, and again to estimate the two missing elements in column $3$. (Since columns $1$ and $4$ are known, they require no calls to confun.)

At times, central differences are used rather than forward differences, in which case twice as many calls to objfun and confun are needed. (The switch to central differences is not under your control.)

If $i<0$ or $i>3$, the default value is used.

This option defines an interval used to estimate derivatives by finite differences in the following circumstances:

(a)	For verifying the objective and/or constraint gradients (see the description of the optional parameter Verify).
(b)	For estimating unspecified elements of the objective and/or constraint Jacobian matrix.

In general, a derivative with respect to the $j$th variable is approximated using the interval ${\delta }_j$, where ${\delta }_j=r\left(1+\left|{\hat{x}}_j\right|\right)$, with $\hat{x}$ the first point feasible with respect to the bounds and linear constraints. If the functions are well scaled, the resulting derivative approximation should be accurate to $\mathit{O}\left(r\right)$. See Gill et al. (1981) for a discussion of the accuracy in finite difference approximations.

If a difference interval is not specified, a finite difference interval will be computed automatically for each variable by a procedure that requires up to six calls of confun and objfun for each element. This option is recommended if the function is badly scaled or you wish to have e04us determine constant elements in the objective and constraint gradients (see the descriptions of confun and objfun in [Parameters]).

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

The scalar $r$ defines the maximum acceptable absolute violations in linear and nonlinear constraints at a ‘feasible’ point; i.e., a constraint is considered satisfied if its violation does not exceed $r$. If $r<\epsilon$ or $r\ge 1$, the default value is used. Using this keyword sets both optional parameters Linear Feasibility Tolerance and Nonlinear Feasibility Tolerance to $r$, if $\epsilon \le r<1$. (Additional details are given under the descriptions of these optional parameters.)

This parameter defines ${\epsilon }_r$, which is intended to be a measure of the accuracy with which the problem functions $F\left(x\right)$ and $c\left(x\right)$ can be computed. If $r<\epsilon$ or $r\ge 1$, the default value is used.

The value of ${\epsilon }_r$ should reflect the relative precision of $1+\left|F\left(x\right)\right|$; i.e., ${\epsilon }_r$ acts as a relative precision when $\left|F\right|$ is large and as an absolute precision when $\left|F\right|$ is small. For example, if $F\left(x\right)$ is typically of order $1000$ and the first six significant digits are known to be correct, an appropriate value for ${\epsilon }_r$ would be ${10}^{-6}$. In contrast, if $F\left(x\right)$ is typically of order ${10}^{-4}$ and the first six significant digits are known to be correct, an appropriate value for ${\epsilon }_r$ would be ${10}^{-10}$. The choice of ${\epsilon }_r$ can be quite complicated for badly scaled problems; see Chapter 8 of Gill et al. (1981) for a discussion of scaling techniques. The default value is appropriate for most simple functions that are computed with full accuracy. However, when the accuracy of the computed function values is known to be significantly worse than full precision, the value of ${\epsilon }_r$ should be large enough so that e04us will not attempt to distinguish between function values that differ by less than the error inherent in the calculation.

This option controls the contents of the upper triangular matrix $R$ (see [Parameters]). e04us works exclusively with the transformed and reordered Hessian $H_Q$, and hence extra computation is required to form the Hessian itself. If $\mathbf{Hessian}='\mathrm{NO}'$, r contains the Cholesky factor of the transformed and reordered Hessian. If $\mathbf{Hessian}='\mathrm{YES}'$, the Cholesky factor of the approximate Hessian itself is formed and stored in r. You should select $\mathbf{Hessian}='\mathrm{YES}'$ if a Warm Start will be used for the next call to e04us.

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 is treated as a step to an unbounded solution. If the change in $x$ during an iteration would exceed the value of $r$, the objective function is considered to be unbounded below in the feasible region. If $r\le 0$, the default value is used.

This option controls the initial value of the upper triangular matrix $R$. If $J$ denotes the objective Jacobian matrix $\nabla f\left(x\right)$, then $J^{\mathrm{T}}J$ is often a good approximation to the objective Hessian matrix ${\nabla }^{2}F\left(x\right)$ (see also optional parameter Reset Frequency).

The value $r$ ($0\le r<1$) controls the accuracy with which the step $\alpha$ taken during each iteration approximates a minimum of the merit function along the search direction (the smaller the value of $r$, the more accurate the linesearch). The default value $r=0.9$ requests an inaccurate search and is appropriate for most problems, particularly those with any nonlinear constraints.

If there are no nonlinear constraints, a more accurate search may be appropriate when it is desirable to reduce the number of major iterations – for example, if the objective function is cheap to evaluate, or if a substantial number of derivatives are unspecified. If $r<0$ or $r\ge 1$, the default value is used.

The default value of $r_2$ is ${\epsilon }^{0.33}$ if $\mathbf{Derivative\; Level}=0$ or $1$, and $\sqrt{\epsilon }$ otherwise.

The scalars $r_1$ and $r_2$ define the maximum acceptable absolute violations in linear and nonlinear constraints at a ‘feasible’ point; i.e., a linear constraint is considered satisfied if its violation does not exceed $r_1$, and similarly for a nonlinear constraint and $r_2$. If $r_m<\epsilon$ or $r_{\mathit{m}}\ge 1$, the default value is used, for $\mathit{m}=1,2$.

On entry to e04us, an iterative procedure is executed in order to find a point that satisfies the linear constraints and bounds on the variables to within the tolerance $r_1$. All subsequent iterates will satisfy the linear constraints to within the same tolerance (unless $r_1$ is comparable to the finite difference interval).

For nonlinear constraints, the feasibility tolerance $r_2$ defines the largest constraint violation that is acceptable at an optimal point. Since nonlinear constraints are generally not satisfied until the final iterate, the value of optional parameter Nonlinear Feasibility Tolerance acts as a partial termination criterion for the iterative sequence generated by e04us (see also optional parameter Optimality Tolerance).

These tolerances should reflect the precision of the corresponding constraints. For example, if the variables and the coefficients in the linear constraints are of order unity, and the latter are correct to about $6$ decimal digits, it would be appropriate to specify $r_1$ as ${10}^{-6}$.

Normally each optional parameter specification is printed as it is supplied. Optional parameter Nolist may be used to suppress the printing and optional parameter List may be used to restore printing.

The value of $i$ specifies the maximum number of major iterations allowed before termination. Setting $i=0$ and $\mathbf{Major\; Print\; Level}>0$ means that the workspace needed will be computed and printed, but no iterations will be performed. If $i<0$, the default value is used.

The value of $i$ controls the amount of printout produced by the major iterations of e04us, as indicated below. A detailed description of the printed output is given in [Description of the Printed Output] (summary output at each major iteration and the final solution) and [Description of Monitoring Information] (monitoring information at each major iteration). (See also the description of the optional parameter Minor Print Level.)

The following printout is sent to the current advisory message unit (as defined by (X04ABF not in this release)):

The following printout is sent to the logical unit number defined by the optional parameter Monitoring File:

If $\mathbf{Major\; Print\; Level}\ge 5$ and the unit number defined by the optional parameter Monitoring File is the same as that defined by (X04ABF not in this release), then the summary output for each major iteration is suppressed.

The value of $i$ specifies the maximum number of iterations for finding a feasible point with respect to the bounds and linear constraints (if any). The value of $i$ also specifies the maximum number of minor iterations for the optimality phase of each QP subproblem. If $i\le 0$, the default value is used.

The value of $i$ controls the amount of printout produced by the minor iterations of e04us (i.e., the iterations of the quadratic programming algorithm), as indicated below. A detailed description of the printed output is given in [Description of the Printed Output] (summary output at each minor iteration and the final QP solution) and [Description of Monitoring Information] (monitoring information at each minor iteration). (See also the description of the optional parameter Major Print Level.)

The following printout is sent to the current advisory message unit (as defined by (X04ABF not in this release)):

The following printout is sent to the logical unit number defined by the optional parameter Monitoring File:

If $\mathbf{Minor\; Print\; Level}\ge 5$ and the unit number defined by the optional parameter Monitoring File is the same as that defined by (X04ABF not in this release), then the summary output for each minor iteration is suppressed.

If $i\ge 0$ and $\mathbf{Major\; Print\; Level}\ge 5$ or $i\ge 0$ and $\mathbf{Minor\; Print\; Level}\ge 5$, monitoring information produced by e04us at every iteration is sent to a file with logical unit number $i$. If $i<0$ and/or $\mathbf{Major\; Print\; Level}<5$ and $\mathbf{Minor\; Print\; Level}<5$, no monitoring information is produced.

The parameter $r$ (${\epsilon }_R\le r<1$) specifies the accuracy to which you wish the final iterate to approximate a solution of the problem. Broadly speaking, $r$ indicates the number of correct figures desired in the objective function at the solution. For example, if $r$ is ${10}^{-6}$ and e04us terminates successfully, the final value of $F$ should have approximately six correct figures. If $r<{\epsilon }_R$ or $r\ge 1$, the default value is used.

e04us will terminate successfully if the iterative sequence of $x$ values is judged to have converged and the final point satisfies the first-order Kuhn–Tucker conditions (see [] in e04uf). The sequence of iterates is considered to have converged at $x$ if where $p$ is the search direction and $\alpha$ the step length. An iterate is considered to satisfy the first-order conditions for a minimum if and where $Z^{\mathrm{T}}{g}_{\mathrm{FR}}$ is the projected gradient, $g_{\mathrm{FR}}$ is the gradient of $F\left(x\right)$ with respect to the free variables, ${\mathit{res}}_j$ is the violation of the $j$th active nonlinear constraint, and $\mathit{ftol}$ is the Nonlinear Feasibility Tolerance.

If $i>0$, this parameter allows you to reset the approximate Hessian matrix to $J^{\mathrm{T}}J$ every $i$ iterations, where $J$ is the objective Jacobian matrix $\nabla f\left(x\right)$ (see also the description of the optional parameter JTJ Initial Hessian).

At any point where there are no nonlinear constraints active and the values of $f$ are small in magnitude compared to the norm of $J$, $J^{\mathrm{T}}J$ will be a good approximation to the objective Hessian ${\nabla }^{2}F\left(x\right)$. Under these circumstances, frequent resetting can significantly improve the convergence rate of e04us.

Resetting is suppressed at any iteration during which there are nonlinear constraints active.

If $i\le 0$, the default value is used.

These keywords take effect only if $\mathbf{Verify\; Level}>0$. They may be used to control the verification of Jacobian elements computed by user-supplied delegates objfun and confun. For example, if the first $30$ columns of the objective Jacobian appeared to be correct in an earlier run, so that only column $31$ remains questionable, it is reasonable to specify $\mathbf{Start\; Objective\; Check\; At\; Variable}=31$. If the first $30$ variables appear linearly in the subfunctions, so that the corresponding Jacobian elements are constant, the above choice would also be appropriate.

If $i_{2\mathit{m}-1}\le 0$ or $i_{2\mathit{m}-1}>\min \left(n,i_{2\mathit{m}}\right)$, the default value is used, for $\mathit{m}=1,2$. If $i_{2\mathit{m}}\le 0$ or $i_{2\mathit{m}}>n$, the default value is used, for $\mathit{m}=1,2$.

If $r>0,r$ specifies the maximum change in variables at the first step of the linesearch. In some cases, such as $F\left(x\right)=a{e}^{bx}$ or $F\left(x\right)=a{x}^b$, even a moderate change in the elements of $x$ can lead to floating-point overflow. The parameter $r$ is therefore used to encourage evaluation of the problem functions at meaningful points. Given any major iterate $x$, the first point $\tilde{x}$ at which $F$ and $c$ are evaluated during the linesearch is restricted so that The linesearch may go on and evaluate $F$ and $c$ at points further from $x$ if this will result in a lower value of the merit function (indicated by L at the end of each line of output produced by the major iterations; see [Description of the Printed Output]). If L is printed for most of the iterations, $r$ should be set to a larger value.

Wherever possible, upper and lower bounds on $x$ should be used to prevent evaluation of nonlinear functions at wild values. The default value $\mathbf{Step\; Limit}=2.0$ should not affect progress on well-behaved functions, but values such as $0.1\text{ or }0.01$ may be helpful when rapidly varying functions are present. If a small value of Step Limit is selected, a good starting point may be required. An important application is to the class of nonlinear least squares problems. If $r\le 0$, the default value is used.

These keywords refer to finite difference checks on the gradient elements computed by objfun and confun. (Unspecified gradient elements are not checked.) The possible choices for $i$ are the following:

For example, the nonlinear objective gradient (if any) will be verified if either Verify Objective Gradients or $\mathbf{Verify\; Level}=1$ is specified. Similarly, the objective and the constraint gradients will be verified if $\mathbf{Verify}='\mathrm{YES}'$ or $\mathbf{Verify\; Level}=3$ or Verify is specified.

If $i=-1$, no checking will be performed.

If $0\le i\le 3$, gradients will be verified at the first point that satisfies the linear constraints and bounds. If $i=0$, only a ‘cheap’ test will be performed, requiring one call to objfun and (if appropriate) one call to confun. If $1\le i\le 3$, a more reliable (but more expensive) check will be made on individual gradient elements, within the ranges specified by the Start Objective Check At Variable and Stop Objective Check At Variable keywords. A result of the form OK or BAD? is printed by e04us to indicate whether or not each element appears to be correct.

If $10\le i\le 13$, the action is the same as for $i-10$, except that it will take place at the user-specified initial value of $x$.

If $i<-1$ or $4\le i\le 9$ or $i>13$, the default value is used.

We suggest that $\mathbf{Verify\; Level}=3$ be used whenever a new function method is being developed.