naginterfaces.library.opt.nlp2_​sparse_​solve

naginterfaces.library.opt.nlp2_sparse_solve(start, objadd, objrow, prob, usrfun, iafun, javar, nea, a, igfun, jgvar, neg, xlow, xupp, xnames, flow, fupp, fnames, x, xstate, f, fstate, fmul, ns, comm, data=None, io_manager=None)[source]

nlp2_sparse_solve solves sparse linear and nonlinear programming problems.

Note: this function uses optional algorithmic parameters, see also: nlp2_sparse_option_file(), nlp2_sparse_option_string(), nlp2_sparse_option_integer_set(), nlp2_sparse_option_double_set(), nlp2_sparse_init(), nlp2_sparse_jacobian().

Deprecated since version 28.3.0.0: nlp2_sparse_solve is deprecated. Please use handle_solve_ssqp() instead. See also the Replacement Calls document.

For full information please refer to the NAG Library document for e04vh

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/e04/e04vhf.html

Parameters
startint

Indicates how a starting point is to be obtained.

Requests that the Crash procedure be used, unless a Basis file is provided via options ‘Old Basis File’, ‘Insert File’ or ‘Load File’.

Is the same as but is more meaningful when a Basis file is given.

Means that and define a valid starting point (probably from an earlier call, though not necessarily).

objaddfloat

Is a constant that will be added to the objective row for printing purposes. Typically, .

objrowint

Says which row of is to act as the objective function. If there is no such row, set . Then nlp2_sparse_solve will seek a feasible point such that and .

probstr, length 8

Is an -character name for the problem. is used in the printed solution and in some functions that output Basis files. A blank name may be used.

usrfuncallable (status, f, g) = usrfun(status, x, needf, f, needg, g, data=None)

must define the nonlinear portion of the problem functions , along with its gradient elements . (A dummy function is needed even if and all functions are linear.)

In general, should return all function and gradient values on every entry except perhaps the last.

This provides maximum reliability and corresponds to the default option setting, .

The elements of are stored in the array in the order specified by the input arrays and .

In practice it is often convenient not to code gradients. nlp2_sparse_solve is able to estimate them by finite differences, using a call to for each variable for which some needs to be estimated.

However, this reduces the reliability of the optimization algorithm, and it can be very expensive if there are many such variables .

As a compromise, nlp2_sparse_solve allows you to code as many gradients as you like.

This option is implemented as follows.

Just before is called, each element of the derivative array is initialized to a specific value.

On exit, any element retaining that value must be estimated by finite differences.

Some rules of thumb follow:

  1. for maximum reliability, compute all gradients;

  2. if the gradients are expensive to compute, specify option ‘Nonderivative Linesearch’ and use the value of the input argument to avoid computing them on certain entries. (There is no need to compute gradients if on entry to .);

  3. if not all gradients are known, you must specify . You should still compute as many gradients as you can. (It often happens that some of them are constant or zero.);

  4. again, if the known gradients are expensive, don’t compute them if on entry to ;

  5. use the input argument to test for special actions on the first or last entries;

  6. while is being developed, use the option ‘Verify Level’ to check the computation of gradients that are supposedly known;

  7. is not called until the linear constraints and bounds on are satisfied. This helps confine to regions where the functions are likely to be defined. However, be aware of the option ‘Minor Feasibility Tolerance’ if the functions have singularities on the constraint boundaries;

  8. set if some of the functions are undefined. The linesearch will shorten the step and try again;

  9. set if you want nlp2_sparse_solve to stop.

Parameters
statusint

Indicates the first and last calls to .

There is nothing special about the current call to .

nlp2_sparse_solve is calling your function for the first time. You may wish to do something special such as read data from a file.

nlp2_sparse_solve is calling your function for the last time. This argument setting allows you to perform some additional computation on the final solution.

The current is optimal.

The problem appears to be infeasible.

The problem appears to be unbounded.

An iterations limit was reached.

If the functions are expensive to evaluate, it may be desirable to do nothing on the last call.

xfloat, ndarray, shape

The variables at which the problem functions are to be calculated. The array must not be altered.

needfint

Indicates whether must be assigned during this call of .

is not required and is ignored.

The components of corresponding to the nonlinear part of must be calculated and assigned to .

If is linear and completely defined by the th row of , , the associated value is ignored and need not be assigned.

However, if has a nonlinear portion that happens to be zero at , it is still necessary to set .

If the linear part of a nonlinear is provided using the arrays , and , it must not be computed again as part of .

To simplify the code, you may ignore the value of and compute on every entry to .

may also be ignored with ‘Derivative Linesearch’ and .

In this case, is always , and must always be assigned.

ffloat, ndarray, shape

Concerns the calculation of .

needgint

Indicates whether must be assigned during this call of .

is not required and is ignored.

The partial derivatives of must be calculated and assigned to . The value of should be , where , and .

gfloat, ndarray, shape

Concerns the calculations of the derivatives of the function .

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
statusint

May be used to indicate that you are unable to evaluate or its gradients at the current . (For example, the problem functions may not be defined there.)

During the linesearch, is evaluated at points for various step lengths , where has already been evaluated satisfactorily.

For any such , if you set , nlp2_sparse_solve will reduce and evaluate again (closer to , where is more likely to be defined).

If for some reason you wish to terminate the current problem, set .

ffloat, array-like, shape

contains the computed functions (except perhaps if ).

gfloat, array-like, shape

Contains the computed derivatives (unless ).

These derivative elements must be stored in in exactly the same positions as implied by the definitions of arrays and .

There is no internal check for consistency (except indirectly via the option ‘Verify Level’), so great care is essential.

iafunint, array-like, shape

and define the coordinates of the nonzero elements of the linear part of the function .

In particular, triples define the row and column indices and of the element .

The coordinates may define the elements of in any order.

javarint, array-like, shape

and define the coordinates of the nonzero elements of the linear part of the function .

In particular, triples define the row and column indices and of the element .

The coordinates may define the elements of in any order.

neaint

Is the number of nonzero entries in such that .

afloat, array-like, shape

Defines the values of the nonzero elements of the linear part of the function .

In particular, triples define the row and column indices and of the element .

The coordinates may define the elements of in any order.

igfunint, array-like, shape

and define the coordinates of the nonzero elements of , the nonlinear part of the derivative of the function . nlp2_sparse_jacobian() may be used to define these two arrays.

The coordinates can define the elements of in any order.

However, must define the actual elements of in exactly the same order as defined by the coordinates .

jgvarint, array-like, shape

and define the coordinates of the nonzero elements of , the nonlinear part of the derivative of the function . nlp2_sparse_jacobian() may be used to define these two arrays.

The coordinates can define the elements of in any order.

However, must define the actual elements of in exactly the same order as defined by the coordinates .

negint

The number of nonzero entries in .

xlowfloat, array-like, shape

Contain the lower and upper bounds and on the variables .

To specify a nonexistent lower bound , set , where is the option ‘Infinite Bound Size’.

To specify a nonexistent upper bound , set .

To fix the th variable at , where , set .

xuppfloat, array-like, shape

Contain the lower and upper bounds and on the variables .

To specify a nonexistent lower bound , set , where is the option ‘Infinite Bound Size’.

To specify a nonexistent upper bound , set .

To fix the th variable at , where , set .

xnamesstr, length 8, array-like, shape

The optional names for the variables.

If , is not referenced and default names will be used for output.

If , should contain the -character name of the th variable.

flowfloat, array-like, shape

Contain the lower and upper bounds and on .

To specify a nonexistent lower bound , set .

For a nonexistent upper bound , set .

To make the th constraint an equality at , where , set .

fuppfloat, array-like, shape

Contain the lower and upper bounds and on .

To specify a nonexistent lower bound , set .

For a nonexistent upper bound , set .

To make the th constraint an equality at , where , set .

fnamesstr, length 8, array-like, shape

The optional names for the problem functions.

If , is not referenced and default names will be used for output.

If , should contain the -character name of the th row of .

xfloat, array-like, shape

An initial estimate of the variables . See the following description of .

xstateint, array-like, shape

The initial state for each variable .

If or and no basis information is provided (the options ‘Old Basis File’, ‘Insert File’ and ‘Load File’ are all set to ; the default) and must be defined.

If nothing special is known about the problem, or if there is no wish to provide special information, you may set , , for all .

If you set set ; if you set , then set .

In this case a Crash procedure is used to select an initial basis.

If or and basis information is provided (at least one of the options ‘Old Basis File’, ‘Insert File’ and ‘Load File’ is nonzero) and need not be set.

If (Warm Start), and must be set (probably from a previous call).

In this case must be , , or , for .

ffloat, array-like, shape

An initial value for the problem functions . See the following description of .

fstateint, array-like, shape

The initial state for the problem functions .

If or and no basis information is provided (the options ‘Old Basis File’, ‘Insert File’ and ‘Load File’ are all set to ; the default, and must be defined.

If nothing special is known about the problem, or if there is no wish to provide special information, you may set , , for all .

Less trivially, to say that the optimal value of function will probably be equal to one of its bounds, set and or and as appropriate.

In this case a Crash procedure is used to select an initial basis.

If or and basis information is provided (at least one of the options ‘Old Basis File’, ‘Insert File’ and ‘Load File’ is nonzero), and need not be set.

If (Warm Start), and must be set (probably from a previous call).

In this case must be , , or , for .

fmulfloat, array-like, shape

An estimate of , the vector of Lagrange multipliers (shadow prices) for the constraints . All components must be defined. If nothing is known about , set , for . For warm start use the values from a previous call.

nsint

The number of superbasic variables. need not be specified for cold starts, but should retain its value from a previous call when warm start is used.

commdict, communication object, modified in place

Communication structure.

This argument must have been initialized by a prior call to nlp2_sparse_init().

dataarbitrary, optional

User-communication data for callback functions.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns
xfloat, ndarray, shape

The final values of the variable .

xstateint, ndarray, shape

The final state of the variables.

State of variable

Usual value of

0

nonbasic

1

nonbasic

2

superbasic

Between and

3

basic

Between and

Basic and superbasic variables may be outside their bounds by as much as the option ‘Minor Feasibility Tolerance’.

Note that if scaling is specified, the feasibility tolerance applies to the variables of the scaled problem.

In this case, the variables of the original problem may be as much as outside their bounds, but this is unlikely unless the problem is very badly scaled.

Check the value of Primal infeasibility output to the unit number (see unit_from_fileobj()) associated with the option ‘Print File’.

Very occasionally some nonbasic variables may be outside their bounds by as much as the option ‘Minor Feasibility Tolerance’, and there may be some nonbasics for which lies strictly between its bounds.

If , some basic and superbasic variables may be outside their bounds by an arbitrary amount (bounded by if scaling was not used).

xmulfloat, ndarray, shape

The vector of the dual variables (Lagrange multipliers) for the simple bounds .

ffloat, ndarray, shape

The final values for the problem functions (the values at the final point ).

fstateint, ndarray, shape

The final state of the variables. The elements of have the following meaning:

State of the corresponding slack variable

Usual value of

0

nonbasic

1

nonbasic

2

superbasic

Between and

3

basic

Between and

Basic and superbasic slack variables may lead to the corresponding functions being outside their bounds by as much as the option ‘Minor Feasibility Tolerance’.

Very occasionally some functions may be outside their bounds by as much as the option ‘Minor Feasibility Tolerance’, and there may be some nonbasics for which lies strictly between its bounds.

If , some basic and superbasic variables may be outside their bounds by an arbitrary amount (bounded by if scaling was not used).

fmulfloat, ndarray, shape

The vector of the dual variables (Lagrange multipliers) for the general constraints

nsint

The final number of superbasic variables.

ninfint

Are the number and the sum of the infeasibilities of constraints that lie outside one of their bounds by more than the option ‘Minor Feasibility Tolerance’ before the solution is unscaled.

If any linear constraints are infeasible, minimizes the sum of the infeasibilities of the linear constraints subject to the upper and lower bounds being satisfied.

In this case gives the number of variables and linear constraints lying outside their upper or lower bounds.

The nonlinear constraints are not evaluated.

Otherwise, minimizes the sum of infeasibilities of the nonlinear constraints subject to the linear constraints and upper and lower bounds being satisfied.

In this case gives the number of components of lying outside their bounds by more than the option ‘Minor Feasibility Tolerance’.

Again this is before the solution is unscaled.

sinffloat

Are the number and the sum of the infeasibilities of constraints that lie outside one of their bounds by more than the option ‘Minor Feasibility Tolerance’ before the solution is unscaled.

If any linear constraints are infeasible, minimizes the sum of the infeasibilities of the linear constraints subject to the upper and lower bounds being satisfied.

In this case gives the number of variables and linear constraints lying outside their upper or lower bounds.

The nonlinear constraints are not evaluated.

Otherwise, minimizes the sum of infeasibilities of the nonlinear constraints subject to the linear constraints and upper and lower bounds being satisfied.

In this case gives the number of components of lying outside their bounds by more than the option ‘Minor Feasibility Tolerance’.

Again this is before the solution is unscaled.

Other Parameters
‘Central Difference Interval’float

Default

When , the central-difference interval is used near an optimal solution to obtain more accurate (but more expensive) estimates of gradients. Twice as many function evaluations are required compared to forward differencing. The interval used for the th variable is . The resulting derivative estimates should be accurate to , unless the functions are badly scaled.

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

‘Check Frequency’int

Default

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

is useful for debugging purposes, but otherwise this option should not be needed.

‘Crash Option’int

Default

Except on restarts, an internal Crash procedure is used to select an initial basis from certain rows and columns of the constraint matrix . The ‘Crash Option’ determines which rows and columns of are eligible initially, and how many times the Crash procedure is called. Columns of are used to pad the basis where necessary.

Meaning

The initial basis contains only slack variables: .

The Crash procedure is called once, looking for a triangular basis in all rows and columns of .

The Crash procedure is called twice (if there are nonlinear constraints). The first call looks for a triangular basis in linear rows, and the iteration proceeds with simplex iterations until the linear constraints are satisfied. The Jacobian is then evaluated for the first major iteration and the Crash procedure is called again to find a triangular basis in the nonlinear rows (retaining the current basis for linear rows).

The Crash procedure is called up to three times (if there are nonlinear constraints). The first two calls treat linear equalities and linear inequalities separately. As before, the last call treats nonlinear rows before the first major iteration.

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

The ‘Crash Tolerance’ allows the starting Crash procedure to ignore certain ‘small’ nonzeros in each column of . If is the largest element in column , other nonzeros of in the columns are ignored if . (To be meaningful, should be in the range .)

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

For example, suppose the first columns of form the matrix shown under ‘LU Factor Tolerance’; i.e., a tridiagonal matrix with entries , , . To help the Crash procedure choose all columns for the initial basis, we would specify a ‘Crash Tolerance’ of for some value of .

‘Crash Tolerance’float

Default

Except on restarts, an internal Crash procedure is used to select an initial basis from certain rows and columns of the constraint matrix . The ‘Crash Option’ determines which rows and columns of are eligible initially, and how many times the Crash procedure is called. Columns of are used to pad the basis where necessary.

Meaning

The initial basis contains only slack variables: .

The Crash procedure is called once, looking for a triangular basis in all rows and columns of .

The Crash procedure is called twice (if there are nonlinear constraints). The first call looks for a triangular basis in linear rows, and the iteration proceeds with simplex iterations until the linear constraints are satisfied. The Jacobian is then evaluated for the first major iteration and the Crash procedure is called again to find a triangular basis in the nonlinear rows (retaining the current basis for linear rows).

The Crash procedure is called up to three times (if there are nonlinear constraints). The first two calls treat linear equalities and linear inequalities separately. As before, the last call treats nonlinear rows before the first major iteration.

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

The ‘Crash Tolerance’ allows the starting Crash procedure to ignore certain ‘small’ nonzeros in each column of . If is the largest element in column , other nonzeros of in the columns are ignored if . (To be meaningful, should be in the range .)

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

For example, suppose the first columns of form the matrix shown under ‘LU Factor Tolerance’; i.e., a tridiagonal matrix with entries , , . To help the Crash procedure choose all columns for the initial basis, we would specify a ‘Crash Tolerance’ of for some value of .

‘Defaults’valueless

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

‘Derivative Option’int

Default

Option ‘Derivative Option’ specifies which nonlinear function gradients are known analytically and will be supplied to nlp2_sparse_solve by .

Meaning

Some problem derivatives are unknown.

All problem derivatives are known.

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

If , nlp2_sparse_solve will estimate the missing components of using finite differences. This may simplify the coding of . However, it could increase the total run-time substantially (since a special call to is required for each column of the Jacobian that has a missing element), and there is less assurance that an acceptable solution will be located. If the nonlinear variables are not well scaled, it may be necessary to specify a nonstandard option ‘Difference Interval’.

For each column of the Jacobian, one call to is needed to estimate all missing elements in that column, if any.

At times, central differences are used rather than forward differences. Twice as many calls to are needed. (This is not under your control.)

‘Derivative Linesearch’valueless

Default

At each major iteration a linesearch is used to improve the merit function. Option ‘Derivative Linesearch’ uses safeguarded cubic interpolation and requires both function and gradient values to compute estimates of the step . If some analytic derivatives are not provided, or option ‘Nonderivative Linesearch’ is specified, nlp2_sparse_solve employs a linesearch based upon safeguarded quadratic interpolation, which does not require gradient evaluations.

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

If ‘Nonderivative Linesearch’ is selected, nlp2_sparse_solve signals the evaluation of the linesearch by calling with . Once the linesearch is completed, the problem functions are called again with and . If the potential saving provided by a nonderivative linesearch is to be realised, it is essential that be coded so that derivatives are not computed when .

‘Nonderivative Linesearch’valueless

At each major iteration a linesearch is used to improve the merit function. Option ‘Derivative Linesearch’ uses safeguarded cubic interpolation and requires both function and gradient values to compute estimates of the step . If some analytic derivatives are not provided, or option ‘Nonderivative Linesearch’ is specified, nlp2_sparse_solve employs a linesearch based upon safeguarded quadratic interpolation, which does not require gradient evaluations.

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

If ‘Nonderivative Linesearch’ is selected, nlp2_sparse_solve signals the evaluation of the linesearch by calling with . Once the linesearch is completed, the problem functions are called again with and . If the potential saving provided by a nonderivative linesearch is to be realised, it is essential that be coded so that derivatives are not computed when .

‘Difference Interval’float

Default

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

  • in the interval (‘cheap’) phase of verifying the problem derivatives;

  • for verifying the problem derivatives;

  • for estimating missing derivatives.

In all cases, a derivative with respect to is estimated by perturbing that component of to the value , and then evaluating or at the perturbed point. The resulting gradient estimates should be accurate to unless the functions are badly scaled. Judicious alteration of may sometimes lead to greater accuracy.

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

‘Dump File’int

Default

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

If , the last solution obtained will be output to the file with unit number .

If , the ‘Load File’, containing basis information, will be read. The file will usually have been output previously as a ‘Dump File’. The file will not be accessed if options ‘Old Basis File’ or ‘Insert File’ are specified.

‘Load File’int

Default

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

If , the last solution obtained will be output to the file with unit number .

If , the ‘Load File’, containing basis information, will be read. The file will usually have been output previously as a ‘Dump File’. The file will not be accessed if options ‘Old Basis File’ or ‘Insert File’ are specified.

‘Elastic Weight’float

Default

This keyword determines the initial weight associated with the problem (1) (see Treatment of Constraint Infeasibilities).

At major iteration , if elastic mode has not yet started, a scale factor is defined from the current objective gradient. Elastic mode is then started if the QP subproblem is infeasible, or the QP dual variables are larger in magnitude than . The QP is resolved in elastic mode with .

Thereafter, major iterations continue in elastic mode until they converge to a point that is optimal for (1) (see Treatment of Constraint Infeasibilities). If the point is feasible for equation (1) , it is declared locally optimal. Otherwise, is increased by a factor of and major iterations continue. If has already reached a maximum allowable value, equation (1) is declared locally infeasible.

‘Expand Frequency’int

Default

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

For linear models, the strategy is to force a positive step at every iteration, at the expense of violating the bounds on the variables by a small amount. Suppose that the option ‘Minor Feasibility Tolerance’ is . Over a period of iterations, the tolerance actually used by nlp2_sparse_solve increases from to (in steps of ).

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

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

‘Factorization Frequency’int

Default

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

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

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

‘Function Precision’float

Default

The relative function precision is intended to be a measure of the relative accuracy with which the nonlinear functions can be computed. For example, if is computed as for some relevant and if the first significant digits are known to be correct, the appropriate value for would be .

Ideally the functions should have magnitude of order . If all functions are substantially less than in magnitude, should be the absolute precision. For example, if at some point and if the first significant digits are known to be correct, the appropriate value for would be .)

The default value of is appropriate for simple analytic functions.

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

‘Hessian Full Memory’valueless

Default if

These options select the method for storing and updating the approximate Hessian. (nlp2_sparse_solve uses a quasi-Newton approximation to the Hessian of the Lagrangian. A BFGS update is applied after each major iteration.)

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

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

‘Hessian Limited Memory’valueless

Default if

These options select the method for storing and updating the approximate Hessian. (nlp2_sparse_solve uses a quasi-Newton approximation to the Hessian of the Lagrangian. A BFGS update is applied after each major iteration.)

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

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

‘Hessian Frequency’int

Default

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

‘Hessian Full Memory’ and have a similar effect to ‘Hessian Limited Memory’ and (except that the latter retains the current diagonal during resets).

‘Hessian Updates’int

Default if ‘Hessian Full Memory’, otherwise

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

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

‘Infinite Bound Size’float

Default

If , defines the ‘infinite’ bound in the definition of the problem constraints. Any upper bound greater than or equal to will be regarded as (and similarly any lower bound less than or equal to will be regarded as ). If , the default value is used.

‘Iterations Limit’int

Default

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

‘Linesearch Tolerance’float

Default

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

The default value requests just moderate accuracy in the linesearch.

If the nonlinear functions are cheap to evaluate, a more accurate search may be appropriate; try .

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

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

‘LU Density Tolerance’float

Default

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

The singularity tolerance, , helps guard against ill-conditioned basis matrices. After is refactorized, the diagonal elements of are tested as follows: if or , the th column of the basis is replaced by the corresponding slack variable. (This is most likely to occur after a restart.)

‘LU Singularity Tolerance’float

Default

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

The singularity tolerance, , helps guard against ill-conditioned basis matrices. After is refactorized, the diagonal elements of are tested as follows: if or , the th column of the basis is replaced by the corresponding slack variable. (This is most likely to occur after a restart.)

‘LU Factor Tolerance’float

Default

The values of and affect the stability of the basis factorization , during refactorization and updates respectively. The lower triangular matrix is a product of matrices of the form

where the multipliers will satisfy . The default values of and usually strike a good compromise between stability and sparsity. They must satisfy , .

For large and relatively dense problems, or (say) may give a useful improvement in stability without impairing sparsity to a serious degree.

For certain very regular structures (e.g., band matrices) it may be necessary to reduce in order to achieve stability. For example, if the columns of include a sub-matrix of the form

one should set both and to values in the range .

‘LU Update Tolerance’float

Default

The values of and affect the stability of the basis factorization , during refactorization and updates respectively. The lower triangular matrix is a product of matrices of the form

where the multipliers will satisfy . The default values of and usually strike a good compromise between stability and sparsity. They must satisfy , .

For large and relatively dense problems, or (say) may give a useful improvement in stability without impairing sparsity to a serious degree.

For certain very regular structures (e.g., band matrices) it may be necessary to reduce in order to achieve stability. For example, if the columns of include a sub-matrix of the form

one should set both and to values in the range .

‘LU Partial Pivoting’valueless

Default

The factorization implements a Markowitz-type search for pivots that locally minimize the fill-in subject to a threshold pivoting stability criterion. The default option is to use threshhold partial pivoting. The options ‘LU Rook Pivoting’ and ‘LU Complete Pivoting’ are more expensive than partial pivoting but are more stable and better at revealing rank, as long as ‘LU Factor Tolerance’ is not too large (say ). When numerical difficulties are encountered, nlp2_sparse_solve automatically reduces the tolerance towards and switches (if necessary) to rook or complete pivoting, before reverting to the default or specified options at the next refactorization (with ‘System Information Yes’, relevant messages are output to the ‘Print File’).

‘LU Complete Pivoting’valueless

The factorization implements a Markowitz-type search for pivots that locally minimize the fill-in subject to a threshold pivoting stability criterion. The default option is to use threshhold partial pivoting. The options ‘LU Rook Pivoting’ and ‘LU Complete Pivoting’ are more expensive than partial pivoting but are more stable and better at revealing rank, as long as ‘LU Factor Tolerance’ is not too large (say ). When numerical difficulties are encountered, nlp2_sparse_solve automatically reduces the tolerance towards and switches (if necessary) to rook or complete pivoting, before reverting to the default or specified options at the next refactorization (with ‘System Information Yes’, relevant messages are output to the ‘Print File’).

‘LU Rook Pivoting’valueless

The factorization implements a Markowitz-type search for pivots that locally minimize the fill-in subject to a threshold pivoting stability criterion. The default option is to use threshhold partial pivoting. The options ‘LU Rook Pivoting’ and ‘LU Complete Pivoting’ are more expensive than partial pivoting but are more stable and better at revealing rank, as long as ‘LU Factor Tolerance’ is not too large (say ). When numerical difficulties are encountered, nlp2_sparse_solve automatically reduces the tolerance towards and switches (if necessary) to rook or complete pivoting, before reverting to the default or specified options at the next refactorization (with ‘System Information Yes’, relevant messages are output to the ‘Print File’).

‘Major Feasibility Tolerance’float

Default

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

Let be the maximum nonlinear constraint violation, normalized by the size of the solution, which is required to satisfy

where is the violation of the th nonlinear constraint, for .

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

‘Major Optimality Tolerance’float

Default

This tolerance, , specifies the final accuracy of the dual variables. On successful termination, nlp2_sparse_solve will have computed a solution such that

where is an estimate of the complementarity slackness for variable , for . The values are computed from the final QP solution using the reduced gradients (where is the th component of the objective gradient, is the associated column of the constraint matrix , and is the set of QP dual variables):

In the ‘Print File’, appears as the quantity labelled ‘Optimal’.

‘Major Iterations Limit’int

Default

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

‘Major Print Level’int

Default

This controls the amount of output to the options ‘Print File’ and ‘Summary File’ at each major iteration. suppresses most output, except for error messages. gives normal output for linear and nonlinear problems, and gives additional details of the Jacobian factorization that commences each major iteration.

In general, the value being specified may be thought of as a binary number of the form

where each letter stands for a digit that is either or as follows:

a single line that gives a summary of each major iteration. (This entry in is not strictly binary since the summary line is printed whenever );

basis statistics, i.e., information relating to the basis matrix whenever it is refactorized. (This output is always provided if );

, the nonlinear variables involved in the objective function or the constraints. These appear under the heading ‘Jacobian variables’;

, the dual variables for the nonlinear constraints. These appear under the heading ‘Multiplier estimates’;

, the values of the nonlinear constraint functions;

, the Jacobian matrix. This appears under the heading ‘ and Jacobian’.

To obtain output of any items , set the corresponding digit to , otherwise to .

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

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

which would mean that is basic at value , and the third column of the Jacobian has elements of and in rows and .

‘Major Step Limit’float

Default

This argument limits the change in during a linesearch. It applies to all nonlinear problems, once a ‘feasible solution’ or ‘feasible subproblem’ has been found.

  1. A linesearch determines a step over the range , where is if there are nonlinear constraints or is the step to the nearest upper or lower bound on if all the constraints are linear. Normally, the first step length tried is .

  2. In some cases, such as or , even a moderate change in the components of can lead to floating-point overflow. The argument is, therefore, used to define a limit (where is the search direction), and the first evaluation of is at the potentially smaller step length .

  3. Wherever possible, upper and lower bounds on should be used to prevent evaluation of nonlinear functions at meaningless points. The option ‘Major Step Limit’ provides an additional safeguard. The default value should not affect progress on well behaved problems, but setting may be helpful when rapidly varying functions are present. A ‘good’ starting point may be required. An important application is to the class of nonlinear least squares problems.

  4. In cases where several local optima exist, specifying a small value for may help locate an optimum near the starting point.

‘Minimize’valueless

Default

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

The keyword ‘Feasible Point’ means ‘Ignore the objective function, while finding a feasible point for the linear and nonlinear constraints’. It can be used to check that the nonlinear constraints are feasible without altering the call to nlp2_sparse_solve.

‘Maximize’valueless

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

The keyword ‘Feasible Point’ means ‘Ignore the objective function, while finding a feasible point for the linear and nonlinear constraints’. It can be used to check that the nonlinear constraints are feasible without altering the call to nlp2_sparse_solve.

‘Feasible Point’valueless

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

The keyword ‘Feasible Point’ means ‘Ignore the objective function, while finding a feasible point for the linear and nonlinear constraints’. It can be used to check that the nonlinear constraints are feasible without altering the call to nlp2_sparse_solve.

‘Minor Feasibility Tolerance’float

Default

nlp2_sparse_solve tries to ensure that all variables eventually satisfy their upper and lower bounds to within this tolerance, . This includes slack variables. Hence, general linear constraints should also be satisfied to within .

Feasibility with respect to nonlinear constraints is judged by the option ‘Major Feasibility Tolerance’ (not by ).

If the bounds and linear constraints cannot be satisfied to within , the problem is declared infeasible. If is quite small, it may be appropriate to raise by a factor of or . Otherwise, some error in the data should be suspected.

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

For example, if , it is essential to place lower bounds on both variables. If , the bounds and might be appropriate. (The log singularity is more serious. In general, keep as far away from singularities as possible.)

If , feasibility is defined in terms of the scaled problem (since it is then more likely to be meaningful).

In reality, nlp2_sparse_solve uses as a feasibility tolerance for satisfying the bounds on and in each QP subproblem. If the sum of infeasibilities cannot be reduced to zero, the QP subproblem is declared infeasible. nlp2_sparse_solve is then in elastic mode thereafter (with only the linearized nonlinear constraints defined to be elastic). See the description of the option ‘Elastic Weight’.

‘Feasibility Tolerance’float

Default

nlp2_sparse_solve tries to ensure that all variables eventually satisfy their upper and lower bounds to within this tolerance, . This includes slack variables. Hence, general linear constraints should also be satisfied to within .

Feasibility with respect to nonlinear constraints is judged by the option ‘Major Feasibility Tolerance’ (not by ).

If the bounds and linear constraints cannot be satisfied to within , the problem is declared infeasible. If is quite small, it may be appropriate to raise by a factor of or . Otherwise, some error in the data should be suspected.

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

For example, if , it is essential to place lower bounds on both variables. If , the bounds and might be appropriate. (The log singularity is more serious. In general, keep as far away from singularities as possible.)

If , feasibility is defined in terms of the scaled problem (since it is then more likely to be meaningful).

In reality, nlp2_sparse_solve uses as a feasibility tolerance for satisfying the bounds on and in each QP subproblem. If the sum of infeasibilities cannot be reduced to zero, the QP subproblem is declared infeasible. nlp2_sparse_solve is then in elastic mode thereafter (with only the linearized nonlinear constraints defined to be elastic). See the description of the option ‘Elastic Weight’.

‘Minor Iterations Limit’int

Default

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

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

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

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

‘Minor Print Level’int

Default

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

Meaning

No minor iteration output except error messages.

A single line of output at each minor iteration (controlled by options ‘Print Frequency’ and ‘Summary Frequency’.

Basis factorization statistics generated during the periodic refactorization of the basis (see the option ‘Factorization Frequency’). Statistics for the first factorization each major iteration are controlled by the option ‘Major Print Level’.

‘New Basis File’int

Default

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

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

Use of is intended as a safeguard against losing the results of a long run. Suppose that a ‘New Basis File’ is being saved every (‘Save Frequency’) iterations, and that nlp2_sparse_solve is about to save such a basis at iteration . It is conceivable that the run may be interrupted during the next few milliseconds (in the middle of the save). In this case the Basis file will be corrupted and the run will have been essentially wasted.

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

Backup Basis File 11
New Basis File 12

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

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

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

‘Backup Basis File’int

Default

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

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

Use of is intended as a safeguard against losing the results of a long run. Suppose that a ‘New Basis File’ is being saved every (‘Save Frequency’) iterations, and that nlp2_sparse_solve is about to save such a basis at iteration . It is conceivable that the run may be interrupted during the next few milliseconds (in the middle of the save). In this case the Basis file will be corrupted and the run will have been essentially wasted.

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

Backup Basis File 11
New Basis File 12

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

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

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

‘Save Frequency’int

Default

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

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

Use of is intended as a safeguard against losing the results of a long run. Suppose that a ‘New Basis File’ is being saved every (‘Save Frequency’) iterations, and that nlp2_sparse_solve is about to save such a basis at iteration . It is conceivable that the run may be interrupted during the next few milliseconds (in the middle of the save). In this case the Basis file will be corrupted and the run will have been essentially wasted.

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

Backup Basis File 11
New Basis File 12

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

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

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

‘New Superbasics Limit’int

Default

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

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

‘Nolist’valueless

Default

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

‘List’valueless

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

‘Old Basis File’int

Default

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

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

‘Partial Price’int

Default

This argument is recommended for large problems that have significantly more variables than constraints. It reduces the work required for each ‘pricing’ operation (where a nonbasic variable is selected to become superbasic). When , all columns of the constraint matrix are searched. Otherwise, and are partitioned to give roughly equal segments and , for . If the previous pricing search was successful on and , the next search begins on the segments and . (All subscripts here are modulo .) If a reduced gradient is found that is larger than some dynamic tolerance, the variable with the largest such reduced gradient (of appropriate sign) is selected to become superbasic. If nothing is found, the search continues on the next segments and , and so on.

For time-stage models having time periods, ‘Partial Price’ (or or ) may be appropriate.

‘Pivot Tolerance’float

Default

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

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

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

‘Print File’int

Default

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

  • a listing of the options;

  • some statistics about the problem;

  • the amount of storage available for the factorization of the basis matrix;

  • notes about the initial basis resulting from a Crash procedure or a Basis file;

  • the iteration log;

  • basis factorization statistics;

  • the exit condition and some statistics about the solution obtained;

  • the printed solution, if requested.

These items are described in Further Comments and Monitoring Information. Further brief output may be directed to the ‘Summary File’.

‘Print Frequency’int

Default

If , one line of the iteration log will be printed every th iteration. A value such as is suggested for those interested only in the final solution. If , the value of is used and effectively no checks are made.

‘Proximal Point Method’int

Default

specifies minimization of or when the starting point is changed to satisfy the linear constraints (where refers to nonlinear variables).

‘Punch File’int

Default

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

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

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

‘Insert File’int

Default

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

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

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

‘Scale Option’int

Default

Three scale options are available as follows:

Meaning

0

No scaling. This is recommended if it is known that and the constraint matrix never have very large elements (say, larger than ).

1

The constraints and variables are scaled by an iterative procedure that attempts to make the matrix coefficients as close as possible to (see Fourer (1982)). This will sometimes improve the performance of the solution procedures.

2

The constraints and variables are scaled by the iterative procedure. Also, a certain additional scaling is performed that may be helpful if the right-hand side or the solution is large. This takes into account columns of that are fixed or have positive lower bounds or negative upper bounds.

Option ‘Scale Tolerance’ affects how many passes might be needed through the constraint matrix. On each pass, the scaling procedure computes the ratio of the largest and smallest nonzero coefficients in each column:

If is less than times its previous value, another scaling pass is performed to adjust the row and column scales. Raising from to (say) usually increases the number of scaling passes through . At most passes are made. The value of should lie in the range .

‘Scale Print’ causes the row scales and column scales to be printed to ‘Print File’, if ‘System Information Yes’ has been specified. The scaled matrix coefficients are , and the scaled bounds on the variables and slacks are , , where if .

‘Scale Tolerance’float

Default

Three scale options are available as follows:

Meaning

0

No scaling. This is recommended if it is known that and the constraint matrix never have very large elements (say, larger than ).

1

The constraints and variables are scaled by an iterative procedure that attempts to make the matrix coefficients as close as possible to (see Fourer (1982)). This will sometimes improve the performance of the solution procedures.

2

The constraints and variables are scaled by the iterative procedure. Also, a certain additional scaling is performed that may be helpful if the right-hand side or the solution is large. This takes into account columns of that are fixed or have positive lower bounds or negative upper bounds.

Option ‘Scale Tolerance’ affects how many passes might be needed through the constraint matrix. On each pass, the scaling procedure computes the ratio of the largest and smallest nonzero coefficients in each column:

If is less than times its previous value, another scaling pass is performed to adjust the row and column scales. Raising from to (say) usually increases the number of scaling passes through . At most passes are made. The value of should lie in the range .

‘Scale Print’ causes the row scales and column scales to be printed to ‘Print File’, if ‘System Information Yes’ has been specified. The scaled matrix coefficients are , and the scaled bounds on the variables and slacks are , , where if .

‘Scale Print’valueless

Three scale options are available as follows:

Meaning

0

No scaling. This is recommended if it is known that and the constraint matrix never have very large elements (say, larger than ).

1

The constraints and variables are scaled by an iterative procedure that attempts to make the matrix coefficients as close as possible to (see Fourer (1982)). This will sometimes improve the performance of the solution procedures.

2

The constraints and variables are scaled by the iterative procedure. Also, a certain additional scaling is performed that may be helpful if the right-hand side or the solution is large. This takes into account columns of that are fixed or have positive lower bounds or negative upper bounds.

Option ‘Scale Tolerance’ affects how many passes might be needed through the constraint matrix. On each pass, the scaling procedure computes the ratio of the largest and smallest nonzero coefficients in each column:

If is less than times its previous value, another scaling pass is performed to adjust the row and column scales. Raising from to (say) usually increases the number of scaling passes through . At most passes are made. The value of should lie in the range .

‘Scale Print’ causes the row scales and column scales to be printed to ‘Print File’, if ‘System Information Yes’ has been specified. The scaled matrix coefficients are , and the scaled bounds on the variables and slacks are , , where if .

‘Solution File’int

Default

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

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

‘Summary File’int

Default

If , a brief log will be output to the file associated with unit , including one line of information every th iteration. In an interactive environment, it is useful to direct this output to the terminal, to allow a run to be monitored online. (If something looks wrong, the run can be manually terminated.) Further details are given in The Summary File.

‘Summary Frequency’int

Default

If , a brief log will be output to the file associated with unit , including one line of information every th iteration. In an interactive environment, it is useful to direct this output to the terminal, to allow a run to be monitored online. (If something looks wrong, the run can be manually terminated.) Further details are given in The Summary File.

‘Superbasics Limit’int

Default

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

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

‘Suppress Parameters’valueless

Normally nlp2_sparse_solve prints the options file as it is being read, and then prints a complete list of the available keywords and their final values. The option ‘Suppress Parameters’ tells nlp2_sparse_solve not to print the full list.

‘System Information No’valueless

Default

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

‘System Information Yes’valueless

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

‘Timing Level’int

Default

If , some timing information will be output to the Print file, if .

‘Unbounded Objective’float

Default

These arguments are intended to detect unboundedness in nonlinear problems. During a linesearch, is evaluated at points of the form , where and are fixed and varies. If exceeds or exceeds , iterations are terminated with the exit message = 5.

If singularities are present, unboundedness in may be manifested by a floating-point overflow (during the evaluation of ), before the test against can be made.

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

‘Unbounded Step Size’float

Default

These arguments are intended to detect unboundedness in nonlinear problems. During a linesearch, is evaluated at points of the form , where and