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.

This option specifies how the initial working set is chosen. With a Cold Start, e04nc 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. e04nc 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 e04nc is called repeatedly to solve related problems.

This value is used in conjunction with the optional parameter Cold Start (the default value) when e04nc 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 $c_j^{\mathrm{T}}x\ge l$ will be included in the initial working set if $\left|c_j^{\mathrm{T}}x-l\right|\le r\left(1+\left|l\right|\right)$. 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.

The scalars $i_1$ and $i_2$ specify the maximum number of iterations allowed in the feasibility and optimality phases. Optional parameter Optimality Phase Iteration Limit is equivalent to optional parameter Iteration Limit. Setting $i_2=0$ 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 $r>\epsilon$, $r$ defines the maximum acceptable absolute violation in each constraint at a ‘feasible’ point. For example, if the variables and the coefficients in the general constaints 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.

Note that a ‘feasible solution’ is a solution that satisfies the current constraints to within the tolerance $r$.

This option controls the contents of the upper triangular matrix $R$ (see the description of a in [Parameters]). e04nc works exclusively with the transformed and reordered matrix $H_Q$ (8), and hence extra computation is required to form the Hessian itself. If $\mathbf{Hessian}='\mathrm{NO}'$, a contains the Cholesky factor of the matrix $H_Q$ with columns ordered as indicated by kx (see [Parameters]). If $\mathbf{Hessian}='\mathrm{YES}'$, a contains the Cholesky factor of the matrix $H$, with columns ordered as indicated by kx.

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 singular and the objective contains an explicit linear term.) 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.

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.

If $i\ge 0$ and $\mathbf{Print\; Level}\ge 5$, monitoring information produced by e04nc 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.

The value of $i$ controls the amount of printout produced by e04nc, 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).

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 optional parameter 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 nine 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 a, b and cvec are not referenced.

The following keywords are also acceptable. The minimum abbreviation of each keyword is underlined.

In addition, the keywords LS and LSQ are equivalent to the default option LS1, and the keyword QP is equivalent to the option QP2.

If $A=0$, i.e., the objective function is purely linear, the efficiency of e04nc may be increased by specifying $a$ as LP.

Note that this option does not apply to problems of type FP or LP.

The default value of $r$ depends on the problem type. If $A$ occurs as a least squares matrix, as it does in problem types QP1, LS1 and LS3, then the default value of $r$ is $100\epsilon$. In all other cases, $A$ is treated as the ‘square root’ of the Hessian matrix $H$ and $r$ has the default value $10\sqrt{\epsilon }$.

This parameter enables you to control the estimate of the triangular factor $R_1$ (see [Main Iteration]). 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 rank of $R$ is defined to be smallest index i such that $\left|R_{i+1,i+1}\right|\le r\left|{\rho }_{i+1}\right|$. If $r\le 0$, the default value is used.