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.

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.

This option specifies how the initial working set is chosen. With a Cold Start, e04nf 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. e04nf 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 e04nf is called repeatedly to solve related problems.

This value is used in conjunction with the optional parameter Cold Start (the default value) when e04nf 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 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 e04nf (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 e04nf 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 $\max \left(i_1,i_2\right)$. 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.

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 ${10}^{-6}$. If $0\le r<\epsilon$, the default value is used.

e04nf 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 $r$.

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.

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.

See optional parameter Feasibility Phase Iteration Limit.

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.

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, $m$.

If no feasible point exists for the constraints then this option is used to control whether or not e04nf will calculate a point that minimizes the constraint violations. If $\mathbf{Minimum\; Sum\; of\; Infeasibilities}='\mathrm{NO}'$, e04nf 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 $\mathbf{Minimum\; Sum\; of\; Infeasibilities}='\mathrm{YES}'$, e04nf will continue until the sum of infeasibilities is minimized.

If $i\ge 0$ and $\mathbf{Print\; Level}\ge 5$, monitoring information produced by e04nf 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.

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.

The value of $i$ controls the amount of printout produced by e04nf, as indicated below. A detailed description of the printed output is given in [Description of the Printed Output] (summary output at each iteration and the final solution) and [Description of Monitoring Information] (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 not in this release)):

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

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

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:

LP	h not referenced, $\mathbf{cvec}\left[\mathbf{n}-1\right]$ required;
QP1	$\mathbf{h}[\mathbf{ldh}-1,\mathrm{!!!!!!!!!}*-1]$ symmetric, cvec not referenced;
QP2	$\mathbf{h}[\mathbf{ldh}-1,\mathrm{!!!!!!!!!}*-1]$ symmetric, $\mathbf{cvec}\left[\mathbf{n}-1\right]$ required;
QP3	$\mathbf{h}[\mathbf{ldh}-1,\mathrm{!!!!!!!!!}*-1]$ upper trapezoidal, cvec not referenced;
QP4	$\mathbf{h}[\mathbf{ldh}-1,\mathrm{!!!!!!!!!}*-1]$ upper trapezoidal, $\mathbf{cvec}\left[\mathbf{n}-1\right]$ required.

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.

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 e04nf may be increased by specifying $a$ as LP.

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 [Algorithmic Details]). If ${\rho }_i$ denotes the function ${\rho }_i=\max \left\{\left|R_{11}\right|,\left|R_{22}\right|,\dots ,\left|R_{ii}\right|\right\}$, 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.