naginterfaces.library.opt.nlp1_​solve

naginterfaces.library.opt.nlp1_solve(a, bl, bu, objfun, x, comm, confun=None, istate=None, cjac=None, clamda=None, r=None, data=None, io_manager=None, spiked_sorder='C')

nlp1_solve is designed to minimize an arbitrary smooth function subject to constraints (which may include simple bounds on the variables, linear constraints and smooth nonlinear constraints) using a Sequential Quadratic Programming (SQP) method. As many first derivatives as possible should be supplied by you; any unspecified derivatives are approximated by finite differences. It is not intended for large sparse problems.

nlp1_solve may also be used for unconstrained, bound-constrained and linearly constrained optimization.

nlp1_solve uses forward communication for evaluating the objective function, the nonlinear constraint functions, and any of their derivatives.

Note: this function uses optional algorithmic parameters, see also: nlp1_option_file(), nlp1_option_string(), nlp1_init().

For full information please refer to the NAG Library document for e04uc

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/e04/e04ucf.html

Parameters

afloat, array-like, shape $(\text{nclin},:)$

Note: the required extent for this argument in dimension 2 is determined as follows: if $\text{nclin}>0$: $n$; otherwise: $1$.

The $\text{i}$th row of $a$ contains the $\text{i}$th row of the matrix $A_L$ of general linear constraints in (1). That is, the $\text{i}$th row contains the coefficients of the $\text{i}$th general linear constraint, for $\text{i}=1,2,\dots ,\text{nclin}$.

blfloat, array-like, shape $(n+\text{nclin}+\text{ncnln})$

$bl$ must contain the lower bounds for all the constraints

bufloat, array-like, shape $(n+\text{nclin}+\text{ncnln})$

$bu$ must contain the upper bounds for all the constraints

objfuncallable (objf, objgrd) = objfun(mode, x, objgrd, nstate, data=None)

$objfun$ must calculate the objective function $F\left(x\right)$ and (optionally) its gradient $g\left(x\right)=\frac{\partial F}{\partial x}$ for a specified $n$-vector $x$.

Parameters

modeint

Indicates which values must be assigned during each call of $objfun$. Only the following values need be assigned:

$mode=0$

$objf$.

$mode=1$

All available elements of $objgrd$.

$mode=2$

$objf$ and all available elements of $objgrd$.

xfloat, ndarray, shape $\left(n\right)$

$x$, the vector of variables at which the objective function and/or all available elements of its gradient are to be evaluated.

objgrdfloat, ndarray, shape $\left(n\right)$

The elements of $objgrd$ are set to special values which enable nlp1_solve to detect whether they are changed by $objfun$.

nstateint

If $nstate=1$, then nlp1_solve is calling $objfun$ for the first time. This argument setting allows you to save computation time if certain data must be read or calculated only once.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

objffloat

If $mode=0$ or $2$, $objf$ must be set to the value of the objective function at $x$.

objgrdfloat, array-like, shape $\left(n\right)$

If $mode=1$ or $2$, $objgrd$ must return the available elements of the gradient evaluated at $x$.

xfloat, array-like, shape $\left(n\right)$

An initial estimate of the solution.

commdict, communication object, modified in place

Communication structure.

This argument must have been initialized by a prior call to nlp1_init().

confunNone or callable (c, cjac) = confun(mode, needc, x, cjac, nstate, data=None), optional

Note: if this argument is None then a NAG-supplied facility will be used.

$confun$ must calculate the vector $c\left(x\right)$ of nonlinear constraint functions and (optionally) its Jacobian ($=\frac{\partial c}{\partial x}$) for a specified $n$-element vector $x$.

If there are no nonlinear constraints (i.e., $\text{ncnln}=0$), $confun$ will never be called by nlp1_solve and $confun$ may be None. If there are nonlinear constraints, the first call to $confun$ will occur before the first call to $objfun$.

Parameters

modeint

Indicates which values must be assigned during each call of $confun$. Only the following values need be assigned, for each value of $i$ such that $needc[i-1]>0$:

$mode=0$

$c[i-1]$.

$mode=1$

All available elements in the $i$th row of $cjac$.

$mode=2$

$c[i-1]$ and all available elements in the $i$th row of $cjac$.

needcint, ndarray, shape $\left(\text{ncnln}\right)$

The indices of the elements of $c$ and/or $cjac$ that must be evaluated by $confun$. If $needc[i-1]>0$, the $i$th element of $c$ and/or the available elements of the $i$th row of $cjac$ (see argument $mode$) must be evaluated at $x$.

xfloat, ndarray, shape $\left(n\right)$

$x$, the vector of variables at which the constraint functions and/or the available elements of the constraint Jacobian are to be evaluated.

cjacfloat, ndarray, shape $(max(1,\text{ncnln}),n)$

Unless $\text{‘Derivative Level'}=2$ or $3$, the elements of $cjac$ are set to special values which enable nlp1_solve to detect whether they are changed by $confun$.

nstateint

If $nstate=1$, then nlp1_solve is calling $confun$ for the first time. This argument setting allows you to save computation time if certain data must be read or calculated only once.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

cfloat, array-like, shape $\left(\text{ncnln}\right)$

If $needc[i-1]>0$ and $mode=0$ or $2$, $c[i-1]$ must contain the value of the $i$th constraint at $x$. The remaining elements of $c$, corresponding to the non-positive elements of $needc$, are ignored.

cjacfloat, array-like, shape $(max(1,\text{ncnln}),n)$

If $needc[i-1]>0$ and $mode=1$ or $2$, the $i$th row of $cjac$ must contain the available elements of the vector $\nabla c_i$ given by

$$\nabla c_i={(\frac{\partial c_i}{\partial x_1},\frac{\partial c_i}{\partial x_2},\dots ,\frac{\partial c_i}{\partial x_n})}^T\text{,}$$

where $\frac{\partial c_i}{\partial x_j}$ is the partial derivative of the $i$th constraint with respect to the $j$th variable, evaluated at the point $x$. See also the argument $nstate$. The remaining rows of $cjac$, corresponding to non-positive elements of $needc$, are ignored.

If all elements of the constraint Jacobian are known (i.e., $\text{‘Derivative Level'}=2$ or $3$), any constant elements may be assigned to $cjac$ one time only at the start of the optimization.

An element of $cjac$ that is not subsequently assigned in $confun$ will retain its initial value throughout.

Constant elements may be loaded into $cjac$ either before the call to nlp1_solve or during the first call to $confun$ (signalled by the value $nstate=1$).

The ability to preload constants is useful when many Jacobian elements are identically zero, in which case $cjac$ may be initialized to zero and nonzero elements may be reset by $confun$.

Note that constant nonzero elements do affect the values of the constraints.

Thus, if $cjac[i-1,j-1]$ is set to a constant value, it need not be reset in subsequent calls to $confun$, but the value $cjac[i-1,j-1]\times x[j-1]$ must nonetheless be added to $c[i-1]$.

For example, if $cjac[0,0]=2$ and $cjac[0,1]=-5$, then the term $2\times x\left[0\right]-5\times x\left[1\right]$ must be included in the definition of $c\left[0\right]$.

It must be emphasized that, if $\text{‘Derivative Level'}=0$ or $1$, unassigned elements of $cjac$ are not treated as constant; they are estimated by finite differences, at nontrivial expense.

If you do not supply a value for the option ‘Difference Interval’, an interval for each element of $x$ is computed automatically at the start of the optimization.

The automatic procedure can usually identify constant elements of $cjac$, which are then computed once only by finite differences.

istateNone or int, array-like, shape $(n+\text{nclin}+\text{ncnln})$, optional

Need not be set if the (default) option ‘Cold Start’ is used.

If the option ‘Warm Start’ has been chosen, the elements of $istate$ corresponding to the bounds and linear constraints define the initial working set for the procedure that finds a feasible point for the linear constraints and bounds.

The active set at the conclusion of this procedure and the elements of $istate$ corresponding to nonlinear constraints then define the initial working set for the first QP subproblem.

More precisely, the first $n$ elements of $istate$ refer to the upper and lower bounds on the variables, the next $n_L$ elements refer to the upper and lower bounds on $A_{L}x$, and the next $n_N$ elements refer to the upper and lower bounds on $c\left(x\right)$.

Possible values for $istate[j-1]$ are as follows:

$istate[j-1]$	Meaning
0	The corresponding constraint is not in the initial QP working set.
1	This inequality constraint should be in the working set at its lower bound.
2	This inequality constraint should be in the working set at its upper bound.
3	This equality constraint should be in the initial working set. This value must not be specified unless $bl[j-1]=bu[j-1]$.

The values $-2$, $-1$ and $4$ are also acceptable but will be modified by the function.

If nlp1_solve has been called previously with the same values of $\text{n}$, $\text{nclin}$ and $\text{ncnln}$, $istate$ already contains satisfactory information.

The function also adjusts (if necessary) the values supplied in $x$ to be consistent with $istate$.

cjacNone or float, array-like, shape $(\text{ncnln},:)$, optional

Note: the required extent for this argument in dimension 2 is determined as follows: if $\text{ncnln}>0$: $n$; otherwise: $1$.

In general, $cjac$ need not be initialized before the call to nlp1_solve. However, if $\text{‘Derivative Level'}=2$ or $3$, you may optionally set the constant elements of $cjac$ (see argument $nstate$ in the description of $confun$). Such constant elements need not be re-assigned on subsequent calls to $confun$.

clamdaNone or float, array-like, shape $(n+\text{nclin}+\text{ncnln})$, optional

Need not be set if the (default) option ‘Cold Start’ is used.

If the option ‘Warm Start’ has been chosen, $clamda[\text{j}-1]$ must contain a multiplier estimate for each nonlinear constraint with a sign that matches the status of the constraint specified by the $istate$ array, for $\text{j}=n+\text{nclin}+1,\dots ,n+\text{nclin}+\text{ncnln}$.

The remaining elements need not be set.

Note that if the $j$th constraint is defined as ‘inactive’ by the initial value of $istate$ array (i.e., $istate[j-1]=0$), $clamda[j-1]$ should be zero; if the $j$th constraint is an inequality active at its lower bound (i.e., $istate[j-1]=1$), $clamda[j-1]$ should be non-negative; if the $j$th constraint is an inequality active at its upper bound (i.e., $istate[j-1]=2$), $clamda[j-1]$ should be non-positive.

If necessary, the function will modify $clamda$ to match these rules.

rNone or float, array-like, shape $(n,n)$, optional

Need not be initialized if the (default) option ‘Cold Start’ is used.

If the option ‘Warm Start’ has been chosen, $r$ must contain the upper triangular Cholesky factor $R$ of the initial approximation of the Hessian of the Lagrangian function, with the variables in the natural order.

Elements not in the upper triangular part of $r$ are assumed to be zero and need not be assigned.

dataarbitrary, optional

User-communication data for callback functions.

io_managerFileObjManager, optional

Manager for I/O in this routine.

spiked_sorderstr, optional

If $a$ and $r$ are spiked (i.e., have unit extent in all but one dimension, or have size $1$), $spiked\_sorder$ selects the storage order to associate with them in the NAG Engine:

spiked_sorder = $\text{'C'}$

row-major storage will be used;

spiked_sorder = $\text{'F'}$

column-major storage will be used.

Two-dimensional arrays returned from callback functions in this routine must then use the same storage order.

Returns

iteraint

The number of major iterations performed.

istateint, ndarray, shape $(n+\text{nclin}+\text{ncnln})$

The status of the constraints in the QP working set at the point returned in $x$. The significance of each possible value of $istate[j-1]$ is as follows:

$istate[j-1]$	Meaning
$-2$	This constraint violates its lower bound by more than the appropriate feasibility tolerance (see the options ‘Linear Feasibility Tolerance’ and ‘Nonlinear Feasibility Tolerance’). This value can occur only when no feasible point can be found for a QP subproblem.
$-1$	This constraint violates its upper bound by more than the appropriate feasibility tolerance (see the options ‘Linear Feasibility Tolerance’ and ‘Nonlinear Feasibility Tolerance’). This value can occur only when no feasible point can be found for a QP subproblem.
$0$	The constraint is satisfied to within the feasibility tolerance, but is not in the QP working set.
$1$	This inequality constraint is included in the QP working set at its lower bound.
$2$	This inequality constraint is included in the QP working set at its upper bound.
$3$	This constraint is included in the QP working set as an equality. This value of $istate$ can occur only when $bl[j-1]=bu[j-1]$.

cNone or float, ndarray, shape $\left(\text{ncnln}\right)$

If $\text{ncnln}>0$, $c[\text{i}-1]$ contains the value of the $\text{i}$th nonlinear constraint function $c_{\text{i}}$ at the final iterate, for $\text{i}=1,2,\dots ,\text{ncnln}$.

If $\text{ncnln}=0$, $c$ is returned as None.

cjacNone or float, ndarray, shape $(\text{ncnln},:)$

If $\text{ncnln}>0$, $cjac$ contains the Jacobian matrix of the nonlinear constraint functions at the final iterate, i.e., $cjac[\text{i}-1,\text{j}-1]$ contains the partial derivative of the $\text{i}$th constraint function with respect to the $\text{j}$th variable, for $\text{j}=1,2,\dots ,n$, for $\text{i}=1,2,\dots ,\text{ncnln}$. (See the discussion of argument $cjac$ under $confun$.)

If $\text{ncnln}=0$, $cjac$ is returned as None.

clamdafloat, ndarray, shape $(n+\text{nclin}+\text{ncnln})$

The values of the QP multipliers from the last QP subproblem. $clamda[j-1]$ should be non-negative if $istate[j-1]=1$ and non-positive if $istate[j-1]=2$.

objffloat

The value of the objective function at the final iterate.

objgrdfloat, ndarray, shape $\left(n\right)$

The gradient of the objective function at the final iterate (or its finite difference approximation).

rfloat, ndarray, shape $(n,n)$

If $\text{‘Hessian'}=\text{'NO'}$, $r$ contains the upper triangular Cholesky factor $R$ of $Q^T\tilde{H}Q$, an estimate of the transformed and reordered Hessian of the Lagrangian at $x$ (see (6) in Overview). If $\text{‘Hessian'}=\text{'YES'}$, $r$ contains the upper triangular Cholesky factor $R$ of $H$, the approximate (untransformed) Hessian of the Lagrangian, with the variables in the natural order.

xfloat, ndarray, shape $\left(n\right)$

The final estimate of the solution.

Other Parameters

‘Central Difference Interval’float

Default values are computed

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 Further Comments). The use of finite differences is discussed further under the option ‘Difference Interval’.

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

‘Cold Start’valueless

Default

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 nlp1_solve 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. nlp1_solve 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 $-2$, $-1$ 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 nlp1_solve is called repeatedly to solve related problems.

‘Warm Start’valueless

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 nlp1_solve 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. nlp1_solve 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 $-2$, $-1$ 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 nlp1_solve is called repeatedly to solve related problems.

‘Crash Tolerance’float

Default $=0.01$

This value is used in conjunction with the option ‘Cold Start’ (the default value) when nlp1_solve 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 $a_j^{T}x\ge l$ will be included in the initial working set if $|a_j^{T}x-l|\le r(1+\left|l\right|)$. If $r<0$ or $r>1$, the default value is used.

‘Defaults’valueless

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

‘Derivative Level’int

Default $=3$

This argument indicates which derivatives are provided in functions $objfun$ and $confun$. The possible choices for $i$ are the following.

$i$	Meaning
3	All elements of the objective gradient and the constraint Jacobian are provided.
2	All elements of the constraint Jacobian are provided, but some elements of the objective gradient are not specified.
1	All elements of the objective gradient are provided, but some elements of the constraint Jacobian are not specified.
0	Some elements of both the objective gradient and the constraint Jacobian are not specified.

The value $i=3$ should be used whenever possible, since nlp1_solve is more reliable (and will usually be more efficient) when all derivatives are exact.

If $i=0$ or $2$, nlp1_solve will estimate the unspecified elements of the objective gradient, 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 Module 8 of Gill et al. (1981), for a discussion of limiting accuracy).

If $i=0$ or $1$, nlp1_solve 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 Jacobian has the form

$$\begin{array}{c}\left(\begin{array}{cccc}\text{*}& \text{*}& \text{*}& \text{*}\\ \text{*}& \text{?}& \text{?}& \text{*}\\ \text{*}& \text{*}& \text{?}& \text{*}\\ \text{*}& \text{*}& \text{*}& \text{*}\end{array}\right)\end{array}$$

where ‘$\ast$’ indicates an element provided by you and ‘?’ indicates an unspecified element, nlp1_solve 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.

‘Difference Interval’float

Default values are computed

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

1. For verifying the objective and/or constraint gradients (see the description of the option ‘Verify’).

2. For estimating unspecified elements of the objective gradient or the constraint Jacobian.

In general, a derivative with respect to the $j$th variable is approximated using the interval ${\delta }_j$, where ${\delta }_j=r(1+\left|{\hat{x}}_j\right|)$, 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 $O\left(r\right)$. See Gill et al. (1981) for a discussion of the accuracy in finite difference approximations.

If a difference interval is not specified by you, 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 nlp1_solve 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 option, a small value between $0.0$ and $1.0$ is appropriate.

‘Feasibility Tolerance’float

Default $=\sqrt{ϵ}$

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 $r<ϵ$ or $r\ge 1$, the default value is used. Using this keyword sets both options ‘Linear Feasibility Tolerance’ and ‘Nonlinear Feasibility Tolerance’ to $r$, if $ϵ\le r<1$. (Additional details are given under the descriptions of these options.)

‘Function Precision’float

Default $={ϵ}^{0.9}$

This argument defines ${ϵ}_r$, 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 $r<ϵ$ or $r\ge 1$, the default value is used.

The value of ${ϵ}_r$ should reflect the relative precision of $1+\left|F\left(x\right)\right|$; i.e., ${ϵ}_r$ 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 ${ϵ}_r$ 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 ${ϵ}_r$ would be ${10}^{-10}$. The choice of ${ϵ}_r$ can be quite complicated for badly scaled problems; see Module 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 ${ϵ}_r$ should be large enough so that nlp1_solve will not attempt to distinguish between function values that differ by less than the error inherent in the calculation.

‘Hessian’str

Default $=NO$

This option controls the contents of the upper triangular matrix $R$ (see Parameters). nlp1_solve works exclusively with the transformed and reordered Hessian $H_Q$ (6), and hence extra computation is required to form the Hessian itself. If $\text{‘Hessian'}=\text{'NO'}$, $r$ contains the Cholesky factor of the transformed and reordered Hessian. If $\text{‘Hessian'}=\text{'YES'}$, the Cholesky factor of the approximate Hessian itself is formed and stored in $r$. You should select $\text{‘Hessian'}=\text{'YES'}$ if a ‘Warm Start’ will be used for the next call to nlp1_solve.

‘Infinite Bound Size’float

Default $={10}^{20}$

If $r>0$, $r$ defines the ‘infinite’ bound $\text{bigbnd}$ in the definition of the problem constraints. Any upper bound greater than or equal to $\text{bigbnd}$ will be regarded as $+\infty$ (and similarly any lower bound less than or equal to $-\text{bigbnd}$ will be regarded as $-\infty$). If $r<0$, the default value is used.

‘Infinite Step Size’float

Default $=max(\text{bigbnd},{10}^{20})$

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.

‘Line Search Tolerance’float

Default $=0.9$

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.

‘Linear Feasibility Tolerance’float

Default $=\sqrt{ϵ}$

The default value of $r_2$ is ${ϵ}^{0.33}$ if $\text{‘Derivative Level'}=0$ or $1$, and $\sqrt{ϵ}$ 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 $r_{\text{m}}<ϵ$ or $r_{\text{m}}\ge 1$, the default value is used, for $\text{m}=1,2,\dots ,2$.

On entry to nlp1_solve, 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 option ‘Nonlinear Feasibility Tolerance’ acts as a partial termination criterion for the iterative sequence generated by nlp1_solve (see the discussion of option ‘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 ${10}^{-6}$.

‘Nonlinear Feasibility Tolerance’float

Default $={ϵ}^{0.33}$ or $\sqrt{ϵ}$

The default value of $r_2$ is ${ϵ}^{0.33}$ if $\text{‘Derivative Level'}=0$ or $1$, and $\sqrt{ϵ}$ 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 $r_{\text{m}}<ϵ$ or $r_{\text{m}}\ge 1$, the default value is used, for $\text{m}=1,2,\dots ,2$.

On entry to nlp1_solve, 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 option ‘Nonlinear Feasibility Tolerance’ acts as a partial termination criterion for the iterative sequence generated by nlp1_solve (see the discussion of option ‘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 ${10}^{-6}$.

‘List’valueless

Option ‘List’ enables printing of each option specification as it is supplied. ‘Nolist’ suppresses this printing.

‘Nolist’valueless

Default

Option ‘List’ enables printing of each option specification as it is supplied. ‘Nolist’ suppresses this printing.

‘Major Iteration Limit’int

Default $=max(50,3(n+n_L)+10n_N)$

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

‘Iteration Limit’int

Default $=max(50,3(n+n_L)+10n_N)$

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

‘Iters’int

Default $=max(50,3(n+n_L)+10n_N)$

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

‘Itns’int

Default $=max(50,3(n+n_L)+10n_N)$

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

‘Major Print Level’int

The value of $i$ controls the amount of printout produced by the major iterations of nlp1_solve, as indicated below. A detailed description of the printed output is given in Further Comments (summary output at each major iteration and the final solution) and Monitoring Information (monitoring information at each major iteration). (See also the description of the option ‘Minor Print Level’.)

The following printout is sent to the file object associated with the advisory I/O unit (see FileObjManager):

$i$	Output
$0$	No output.
$1$	The final solution only.
$5$	One line of summary output ($<80$ characters; see Further Comments) for each major iteration (no printout of the final solution).
$\ge 10$	The final solution and one line of summary output for each major iteration.

The following printout is sent to the unit number given by the option ‘Monitoring File’:

$i$	Output
$<5$	No output.
$\ge 5$	One long line of output ($>80$ characters; see Monitoring Information) for each major iteration (no printout of the final solution).
$\ge 20$	At each major iteration, the objective function, the Euclidean norm of the nonlinear constraint violations, the values of the nonlinear constraints (the vector $c$), the values of the linear constraints (the vector $A_{L}x$), and the current values of the variables (the vector $x$).
$\ge 30$	At each major iteration, the diagonal elements of the matrix $T$ associated with the $TQ$ factorization (5) (see Overview) of the QP working set, and the diagonal elements of $R$, the triangular factor of the transformed and reordered Hessian (6) (see Overview).

If $\text{‘Major Print Level'}\ge 5$ and the unit number defined by the option ‘Monitoring File’ is the advisory unit number, the summary output for each major iteration is suppressed.

‘Print Level’int

Default for nlp1_solve $=0$

The value of $i$ controls the amount of printout produced by the major iterations of nlp1_solve, as indicated below. A detailed description of the printed output is given in Further Comments (summary output at each major iteration and the final solution) and Monitoring Information (monitoring information at each major iteration). (See also the description of the option ‘Minor Print Level’.)

The following printout is sent to the file object associated with the advisory I/O unit (see FileObjManager):

$i$	Output
$0$	No output.
$1$	The final solution only.
$5$	One line of summary output ($<80$ characters; see Further Comments) for each major iteration (no printout of the final solution).
$\ge 10$	The final solution and one line of summary output for each major iteration.

The following printout is sent to the unit number given by the option ‘Monitoring File’:

$i$	Output
$<5$	No output.
$\ge 5$	One long line of output ($>80$ characters; see Monitoring Information) for each major iteration (no printout of the final solution).
$\ge 20$	At each major iteration, the objective function, the Euclidean norm of the nonlinear constraint violations, the values of the nonlinear constraints (the vector $c$), the values of the linear constraints (the vector $A_{L}x$), and the current values of the variables (the vector $x$).
$\ge 30$	At each major iteration, the diagonal elements of the matrix $T$ associated with the $TQ$ factorization (5) (see Overview) of the QP working set, and the diagonal elements of $R$, the triangular factor of the transformed and reordered Hessian (6) (see Overview).

If $\text{‘Major Print Level'}\ge 5$ and the unit number defined by the option ‘Monitoring File’ is the advisory unit number, the summary output for each major iteration is suppressed.

‘Minor Iteration Limit’int

Default $=max(50,3(n+n_L+n_N))$

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.

‘Minor Print Level’int

Default $=0$

The value of $i$ controls the amount of printout produced by the minor iterations of nlp1_solve (i.e., the iterations of the quadratic programming algorithm), as indicated below. A detailed description of the printed output is given in Further Comments (summary output at each minor iteration and the final QP solution) and Monitoring Information (monitoring information at each minor iteration). (See also the description of the option ‘Major Print Level’.)

The following printout is sent to the file object associated with the advisory I/O unit (see FileObjManager):

$i$	Output
$0$	No output.
$1$	The final QP solution only.
$5$	One line of summary output ($<80$ characters; see Further Comments) for each minor iteration (no printout of the final QP solution).
$\ge 10$	The final QP solution and one line of summary output for each minor iteration.

The following printout is sent to the unit number given by the option ‘Monitoring File’:

$i$	Output
$<5$	No output.
$\ge 5$	One long line of output ($>80$ characters; see Further Comments) for each minor iteration (no printout of the final QP solution).
$\ge 20$	At each minor iteration, the current estimates of the QP multipliers, the current estimate of the QP search direction, the QP constraint values, and the status of each QP constraint.
$\ge 30$	At each minor iteration, the diagonal elements of the matrix $T$ associated with the $TQ$ factorization (5) (see Overview) of the QP working set, and the diagonal elements of the Cholesky factor $R$ of the transformed Hessian (6) (see Overview).

If $\text{‘Major Print Level'}\ge 5$ and the unit number defined by the option ‘Monitoring File’ is the advisory unit number, the summary output for each major iteration is suppressed.

‘Monitoring File’int

Default $=-1$

If $i\ge 0$ and $\text{‘Major Print Level'}\ge 5$ or $i\ge 0$ and $\text{‘Minor Print Level'}\ge 5$, monitoring information produced by nlp1_solve at every iteration is sent to a file with logical unit number $i$. If $i<0$ and/or $\text{‘Major Print Level'}<5$ and $\text{‘Minor Print Level'}<5$, no monitoring information is produced.

‘Optimality Tolerance’float

Default $={ϵ}_r^{0.8}$

The argument $r$ (${ϵ}_r\le r<1$) 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 nlp1_solve terminates successfully, the final value of $F$ should have approximately six correct figures. If $r<{ϵ}_r$ or $r\ge 1$, the default value is used.

nlp1_solve 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 Overview). The sequence of iterates is considered to have converged at $x$ if

$$\alpha \parallel p\parallel \le \sqrt{r}(1+\parallel x\parallel )\text{,}$$

where $p$ is the search direction and $\alpha$ the step length from (3). An iterate is considered to satisfy the first-order conditions for a minimum if

$$\Vert Z^T{g}_{FR}\Vert \le \sqrt{r}(1+max(1+\left|F\left(x\right)\right|,\parallel g_{FR}\parallel ))$$

and

$$\left|{\text{res}}_j\right|\le \text{ftol}\text{for all}j\text{,}$$

where $Z^T{g}_{FR}$ is the projected gradient (see Overview), $g_{FR}$ is the gradient of $F\left(x\right)$ with respect to the free variables, ${\text{res}}_j$ is the violation of the $j$th active nonlinear constraint, and $\text{ftol}$ is the ‘Nonlinear Feasibility Tolerance’.

‘Start Objective Check At Variable’int

Default $=1$

These keywords take effect only if $\text{‘Verify Level'}>0$. They may be used to control the verification of gradient elements computed by $objfun$ and/or Jacobian elements computed by $confun$. For example, 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 $\text{‘Start Objective Check At Variable'}=31$. 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 $i_{2\text{m}-1}\le 0$ or $i_{2\text{m}-1}>min(n,i_{2\text{m}})$, the default value is used, for $\text{m}=1,2,\dots ,2$. If $i_{2\text{m}}\le 0$ or $i_{2\text{m}}>n$, the default value is used, for $\text{m}=1,2,\dots ,2$.

‘Stop Objective Check At Variable’int

Default $=n$

These keywords take effect only if $\text{‘Verify Level'}>0$. They may be used to control the verification of gradient elements computed by $objfun$ and/or Jacobian elements computed by $confun$. For example, 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 $\text{‘Start Objective Check At Variable'}=31$. 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 $i_{2\text{m}-1}\le 0$ or $i_{2\text{m}-1}>min(n,i_{2\text{m}})$, the default value is used, for $\text{m}=1,2,\dots ,2$. If $i_{2\text{m}}\le 0$ or $i_{2\text{m}}>n$, the default value is used, for $\text{m}=1,2,\dots ,2$.

‘Start Constraint Check At Variable’int

Default $=1$

These keywords take effect only if $\text{‘Verify Level'}>0$. They may be used to control the verification of gradient elements computed by $objfun$ and/or Jacobian elements computed by $confun$. For example, 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 $\text{‘Start Objective Check At Variable'}=31$. 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 $i_{2\text{m}-1}\le 0$ or $i_{2\text{m}-1}>min(n,i_{2\text{m}})$, the default value is used, for $\text{m}=1,2,\dots ,2$. If $i_{2\text{m}}\le 0$ or $i_{2\text{m}}>n$, the default value is used, for $\text{m}=1,2,\dots ,2$.

‘Stop Constraint Check At Variable’int

Default $=n$

These keywords take effect only if $\text{‘Verify Level'}>0$. They may be used to control the verification of gradient elements computed by $objfun$ and/or Jacobian elements computed by $confun$. For example, 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 $\text{‘Start Objective Check At Variable'}=31$. 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 $i_{2\text{m}-1}\le 0$ or $i_{2\text{m}-1}>min(n,i_{2\text{m}})$, the default value is used, for $\text{m}=1,2,\dots ,2$. If $i_{2\text{m}}\le 0$ or $i_{2\text{m}}>n$, the default value is used, for $\text{m}=1,2,\dots ,2$.

‘Step Limit’float

Default $=2.0$

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 argument $r$ 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

$${\parallel \tilde{x}-x\parallel }_2\le r(1+{\parallel x\parallel }_2)\text{.}$$

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 Further Comments). 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 $\text{‘Step Limit'}=2.0$ should not affect progress on well-behaved functions, but values such as $0.1$ 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.

‘Verify Level’int

Default $=0$

These keywords refer to finite difference checks on the gradient elements computed by $objfun$ and $confun$. The possible choices for $i$ are as follows:

$i$	Meaning
$-1$	No checks are performed.
$0$	Only a ‘cheap’ test will be performed.
$\ge 1$	Individual gradient elements will also be checked using a reliable (but more expensive) test.

It is possible to specify $\text{‘Verify Level'}=0$ to $3$ in several ways. For example, the nonlinear objective gradient (if any) will be verified if either ‘Verify Objective Gradients’ or $\text{‘Verify Level'}=1$ is specified. The constraint gradients will be verified if $\text{‘Verify'}=\text{'YES'}$ or $\text{‘Verify Level'}=2$ or ‘Verify’ is specified. Similarly, the objective and the constraint gradients will be verified if $\text{‘Verify'}=\text{'YES'}$ or $\text{‘Verify Level'}=3$ or ‘Verify’ is specified.

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 Constraint Check At Variable’ and ‘Stop Constraint Check At Variable’ keywords. A result of the form OK or BAD? is printed by nlp1_solve to indicate whether or not each element appears to be correct. If a gradient element is determined to be extremely poor (i.e., if it appears to have no significant digits of accuracy at all), then nlp1_solve will also exit with an error indicator in argument $\text{errno}$.

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 $\text{‘Verify Level'}=3$ be used whenever a new function function is being developed.

‘Verify’valueless

These keywords refer to finite difference checks on the gradient elements computed by $objfun$ and $confun$. The possible choices for $i$ are as follows:

$i$	Meaning
$-1$	No checks are performed.
$0$	Only a ‘cheap’ test will be performed.
$\ge 1$	Individual gradient elements will also be checked using a reliable (but more expensive) test.

It is possible to specify $\text{‘Verify Level'}=0$ to $3$ in several ways. For example, the nonlinear objective gradient (if any) will be verified if either ‘Verify Objective Gradients’ or $\text{‘Verify Level'}=1$ is specified. The constraint gradients will be verified if $\text{‘Verify'}=\text{'YES'}$ or $\text{‘Verify Level'}=2$ or ‘Verify’ is specified. Similarly, the objective and the constraint gradients will be verified if $\text{‘Verify'}=\text{'YES'}$ or $\text{‘Verify Level'}=3$ or ‘Verify’ is specified.

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 Constraint Check At Variable’ and ‘Stop Constraint Check At Variable’ keywords. A result of the form OK or BAD? is printed by nlp1_solve to indicate whether or not each element appears to be correct. If a gradient element is determined to be extremely poor (i.e., if it appears to have no significant digits of accuracy at all), then nlp1_solve will also exit with an error indicator in argument $\text{errno}$.

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 $\text{‘Verify Level'}=3$ be used whenever a new function function is being developed.

‘Verify Constraint Gradients’valueless

These keywords refer to finite difference checks on the gradient elements computed by $objfun$ and $confun$. The possible choices for $i$ are as follows:

$i$	Meaning
$-1$	No checks are performed.
$0$	Only a ‘cheap’ test will be performed.
$\ge 1$	Individual gradient elements will also be checked using a reliable (but more expensive) test.

It is possible to specify $\text{‘Verify Level'}=0$ to $3$ in several ways. For example, the nonlinear objective gradient (if any) will be verified if either ‘Verify Objective Gradients’ or $\text{‘Verify Level'}=1$ is specified. The constraint gradients will be verified if $\text{‘Verify'}=\text{'YES'}$ or $\text{‘Verify Level'}=2$ or ‘Verify’ is specified. Similarly, the objective and the constraint gradients will be verified if $\text{‘Verify'}=\text{'YES'}$ or $\text{‘Verify Level'}=3$ or ‘Verify’ is specified.

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 Constraint Check At Variable’ and ‘Stop Constraint Check At Variable’ keywords. A result of the form OK or BAD? is printed by nlp1_solve to indicate whether or not each element appears to be correct. If a gradient element is determined to be extremely poor (i.e., if it appears to have no significant digits of accuracy at all), then nlp1_solve will also exit with an error indicator in argument $\text{errno}$.

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 $\text{‘Verify Level'}=3$ be used whenever a new function function is being developed.

‘Verify Gradients’valueless

These keywords refer to finite difference checks on the gradient elements computed by $objfun$ and $confun$. The possible choices for $i$ are as follows:

$i$	Meaning
$-1$	No checks are performed.
$0$	Only a ‘cheap’ test will be performed.
$\ge 1$	Individual gradient elements will also be checked using a reliable (but more expensive) test.

It is possible to specify $\text{‘Verify Level'}=0$ to $3$ in several ways. For example, the nonlinear objective gradient (if any) will be verified if either ‘Verify Objective Gradients’ or $\text{‘Verify Level'}=1$ is specified. The constraint gradients will be verified if $\text{‘Verify'}=\text{'YES'}$ or $\text{‘Verify Level'}=2$ or ‘Verify’ is specified. Similarly, the objective and the constraint gradients will be verified if $\text{‘Verify'}=\text{'YES'}$ or $\text{‘Verify Level'}=3$ or ‘Verify’ is specified.

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 Constraint Check At Variable’ and ‘Stop Constraint Check At Variable’ keywords. A result of the form OK or BAD? is printed by nlp1_solve to indicate whether or not each element appears to be correct. If a gradient element is determined to be extremely poor (i.e., if it appears to have no significant digits of accuracy at all), then nlp1_solve will also exit with an error indicator in argument $\text{errno}$.

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 $\text{‘Verify Level'}=3$ be used whenever a new function function is being developed.

‘Verify Objective Gradients’valueless

These keywords refer to finite difference checks on the gradient elements computed by $objfun$ and $confun$. The possible choices for $i$ are as follows:

$i$	Meaning
$-1$	No checks are performed.
$0$	Only a ‘cheap’ test will be performed.
$\ge 1$	Individual gradient elements will also be checked using a reliable (but more expensive) test.

It is possible to specify $\text{‘Verify Level'}=0$ to $3$ in several ways. For example, the nonlinear objective gradient (if any) will be verified if either ‘Verify Objective Gradients’ or $\text{‘Verify Level'}=1$ is specified. The constraint gradients will be verified if $\text{‘Verify'}=\text{'YES'}$ or $\text{‘Verify Level'}=2$ or ‘Verify’ is specified. Similarly, the objective and the constraint gradients will be verified if $\text{‘Verify'}=\text{'YES'}$ or $\text{‘Verify Level'}=3$ or ‘Verify’ is specified.

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 Constraint Check At Variable’ and ‘Stop Constraint Check At Variable’ keywords. A result of the form OK or BAD? is printed by nlp1_solve to indicate whether or not each element appears to be correct. If a gradient element is determined to be extremely poor (i.e., if it appears to have no significant digits of accuracy at all), then nlp1_solve will also exit with an error indicator in argument $\text{errno}$.

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 $\text{‘Verify Level'}=3$ be used whenever a new function function is being developed.

Raises

NagValueError

(errno $7$)

Large errors found in the derivatives.

(errno $9$)

On entry, $n=⟨value⟩$.

Constraint: $n>0$.

(errno $9$)

On entry, $\text{nclin}=⟨value⟩$.

Constraint: $\text{nclin}\ge 0$.

(errno $9$)

On entry, $\text{ncnln}=⟨value⟩$.

Constraint: $\text{ncnln}\ge 0$.

(errno $9$)

Not enough workspace to solve problem. Workspace provided is $\text{iwork}[⟨value⟩]$ and $\text{work}[⟨value⟩]$. To solve problem we need $\text{iwork}[⟨value⟩]$ and $\text{work}[⟨value⟩]$.

(errno $9$)

On entry, the equal bounds on $⟨value⟩$ are infinite, because $bl[⟨value⟩]=\text{beta}$ and $bu[⟨value⟩]=\text{beta}$, but $\left|\text{beta}\right|\ge \text{bigbnd}$: $\text{beta}=⟨value⟩$ and $\text{bigbnd}=⟨value⟩$.

(errno $9$)

On entry, the bounds on $⟨value⟩$ are inconsistent: $bl[⟨value⟩]=⟨value⟩$ and $bu[⟨value⟩]=⟨value⟩$.

(errno $9$)

On entry with a Warm Start, $istate[⟨value⟩]=⟨value⟩$.

(errno $9$)

On entry, the equal bounds on variable $⟨value⟩$ are infinite, because $bl[⟨value⟩]=\text{beta}$ and $bu[⟨value⟩]=\text{beta}$, but $\left|\text{beta}\right|\ge \text{bigbnd}$: $\text{beta}=⟨value⟩$ and $\text{bigbnd}=⟨value⟩$.

(errno $9$)

On entry, the equal bounds on linear constraint $⟨value⟩$ are infinite, because $bl[⟨value⟩]=\text{beta}$ and $bu[⟨value⟩]=\text{beta}$, but $\left|\text{beta}\right|\ge \text{bigbnd}$: $\text{beta}=⟨value⟩$ and $\text{bigbnd}=⟨value⟩$.

(errno $9$)

On entry, the equal bounds on nonlinear constraint $⟨value⟩$ are infinite, because $bl[⟨value⟩]=\text{beta}$ and $bu[⟨value⟩]=\text{beta}$, but $\left|\text{beta}\right|\ge \text{bigbnd}$: $\text{beta}=⟨value⟩$ and $\text{bigbnd}=⟨value⟩$.

(errno $9$)

On entry, the bounds on variable $⟨value⟩$ are inconsistent: $bl[⟨value⟩]=⟨value⟩$ and $bu[⟨value⟩]=⟨value⟩$.

(errno $9$)

On entry, the bounds on linear constraint $⟨value⟩$ are inconsistent: $bl[⟨value⟩]=⟨value⟩$ and $bu[⟨value⟩]=⟨value⟩$.

(errno $9$)

On entry, the bounds on nonlinear constraint $⟨value⟩$ are inconsistent: $bl[⟨value⟩]=⟨value⟩$ and $bu[⟨value⟩]=⟨value⟩$.

Warns

NagAlgorithmicWarning

(errno $1$)

Optimal solution found, but requested accuracy not achieved.

(errno $2$)

No feasible point for the linear constraints.

(errno $3$)

No feasible point for the nonlinear constraints.

(errno $6$)

Current point cannot be improved upon.

NagAlgorithmicMajorWarning

(errno $4$)

Too many major iterations.

NagCallbackTerminateWarning

(errno $i<0$)

User requested termination by setting $mode$ negative in $objfun$ or $confun$.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

nlp1_solve is designed to solve the nonlinear programming problem – the minimization of a smooth nonlinear function subject to a set of constraints on the variables. The problem is assumed to be stated in the following form:

$$\begin{array}{c}{\text{minimize}}_{x\in R^n}F\left(x\right)\text{subject to}l\le \left(\begin{array}{c}x\\ A_{L}x\\ c\left(x\right)\end{array}\right)\le u\text{,}\end{array}$$

where $F\left(x\right)$ (the objective function) is a nonlinear function, $A_L$ is an $n_L\times n$ constant matrix, and $c\left(x\right)$ is an $n_N$ element vector of nonlinear constraint functions. (The matrix $A_L$ and the vector $c\left(x\right)$ may be empty.) The objective function and the constraint functions are assumed to be smooth, i.e., at least twice-continuously differentiable. (The method of nlp1_solve will usually solve (1) if there are only isolated discontinuities away from the solution.)

Note that although the bounds on the variables could be included in the definition of the linear constraints, we prefer to distinguish between them for reasons of computational efficiency. For the same reason, the linear constraints should not be included in the definition of the nonlinear constraints. Upper and lower bounds are specified for all the variables and for all the 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 option ‘Infinite Bound Size’.)

If there are no nonlinear constraints in (1) and $F$ is linear or quadratic, then it will generally be more efficient to use one of lp_solve(), lsq_lincon_solve() or qp_dense_solve(), or handle_solve_ssqp() if the problem is large and sparse. If the problem is large and sparse and does have nonlinear constraints, then handle_solve_ssqp() should be used, since nlp1_solve treats all matrices as dense.

You must supply an initial estimate of the solution to (1), together with functions that define $F\left(x\right)$, $c\left(x\right)$ and as many first partial derivatives as possible; unspecified derivatives are approximated by finite differences.

The objective function is defined by $objfun$, and the nonlinear constraints are defined by $confun$. On every call, these functions must return appropriate values of the objective and nonlinear constraints. You should also provide the available partial derivatives. Any unspecified derivatives are approximated by finite differences; see Other Parameters for a discussion of the option ‘Derivative Level’. Note that if there are any nonlinear constraints then the first call to $confun$ will precede the first call to $objfun$.

For maximum reliability, it is preferable for you to provide all partial derivatives (see Module 8 of Gill et al. (1981), for a detailed discussion). If all gradients cannot be provided, it is similarly advisable to provide as many as possible. While developing $objfun$ and $confun$, the option ‘Verify’ should be used to check the calculation of any known gradients.

The method used by nlp1_solve is described in detail in Algorithmic Details.

nlp1_rcomm() is an alternative function which uses exactly the same method, but uses reverse communication for evaluating the objective and constraint functions.

References

Dennis, J E Jr and Moré, J J, 1977, Quasi-Newton methods, motivation and theory, SIAM Rev. (19), 46–89

Dennis, J E Jr and Schnabel, R B, 1981, A new derivation of symmetric positive-definite secant updates, nonlinear programming, (eds O L Mangasarian, R R Meyer and S M Robinson) (4), 167–199, Academic Press

Dennis, J E Jr and Schnabel, R B, 1983, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice–Hall

Fletcher, R, 1987, Practical Methods of Optimization, (2nd Edition), Wiley

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, Murray, W, Saunders, M A and Wright, M H, 1984, Users’ guide for SOL/QPSOL version 3.2, Report SOL, 84–5, Department of Operations Research, Stanford University

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. Software (10), 282–298

Gill, P E, Murray, W, Saunders, M A and Wright, M H, 1986, Some theoretical properties of an augmented Lagrangian merit function, Report SOL, 86–6R, Department of Operations Research, Stanford University

Gill, P E, Murray, W, Saunders, M A and Wright, M H, 1986, Users’ guide for NPSOL (Version 4.0): a Fortran package for nonlinear programming, Report SOL 86-2, Department of Operations Research, Stanford University

Gill, P E, Murray, W and Wright, M H, 1981, Practical Optimization, Academic Press

Hock, W and Schittkowski, K, 1981, Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems (187), Springer–Verlag

Powell, M J D, 1974, Introduction to constrained optimization, Numerical Methods for Constrained Optimization, (eds P E Gill and W Murray), 1–28, Academic Press

Powell, M J D, 1983, Variable metric methods in constrained optimization, Mathematical Programming: the State of the Art, (eds A Bachem, M Grötschel and B Korte), 288–311, Springer–Verlag

See also

naginterfaces.library.examples.opt.nlp1_solve_ex.main()