SNOPT
Description
SNOPT is a collection of FORTRAN 77 subroutines for solving nonlinear programming problems expressed in the following form:
|
Fobj(x) |
|
L <= F(x) <= U |
l <= x <= u |
where
x is the vector of variables,
l and u are vectors of lower and upper bounds on the variables,
Fobj is a differentiable (nonlinear) function,
F is a vector of differentiable nonlinear functions, and
L and U are vectors of lower and upper bounds on F.
Some of the lower bounds may be – ¥ and some of the upper bounds may be + ¥. The scalar value n will denote the number of variables and m the number of constraints (i.e., the size of vector F) when they appear in this topic.
To solve nonlinear programming problems, SNOPT applies a sparse sequential quadratic programming (SQP) method, using quasi-Newton approximations to the Hessian of the Lagrangian. A brief description of the main features of the SQP algorithm used in SNOPT can be found below.
SNOPT is suitable for nonlinear problems with thousands of constraints and variables, but not thousands of degrees of freedom. (Thus, for large problems there should be many constraints and bounds, and many of them should be active at a solution.)
The relationship between the model formulation above and the AIMMS model is simple, the inequalities defined in AIMMS with <= or >= are converted into equalities by the addition of properly bounded slacks. Slacks with lower and upper bounds of zero are added to all AIMMS equalities to ensure that the Jacobian matrix - the matrix of derivatives of the functions in g with respect to x - has full row rank. All these slacks are, together with the normal AIMMS variables, included in x. The vector l represents the lower bounds as defined in AIMMS, either with the range specified at declaration, or explicitly with the ‘.lower’ qualifier, and any bounds on the slacks. Similarly, u represents upper bounds as defined in AIMMS and any bounds on the slacks. The function g represents the non-constant terms of the AIMMS equations themselves; non-constant terms appearing on the right-hand side are moved to the left-hand side.
SNOPT is equipped with parameters that influence the performance of SNOPT. AIMMS is equipped with options that set the parameters in SNOPT. Options in AIMMS can be set in the options dialog box.
The default values of the parameters are chosen to provide maximum robustness. These choices often cause SNOPT to take more computer time on easy-to-solve problems. In many cases it is possible to reduce the time needed to solve a problem by carefully choosing the parameters to match the problem being solved.
Two parameters that often influence run time are the Calling Crash Procedure option and the Hessian Updates option. The best values for these parameters will vary with each problem, but the following values often make problems with easy-to-moderate difficulty solve more quickly: Calling Crash Procedure equal to ‘Call Crash three times’ and Hessian Updates equal to 5.
Note that the AIMMS general solvers option Iteration Limit defines (in case of SNOPT) a limit on the number of major iterations. It is intended to guard against an excessive number of linearizations of the constraints.
Presolving
AIMMS contains a nonlinear presolve algorithm with the goal to reduce the size of the problem and to tighten the variable bounds, which may help SNOPT to solve nonlinear problems faster. SNOPT is a local solver, i.e., the solution found by the SNOPT is a local solution and cannot be guaranteed to be a global solution. The presolve algorithm may help the solver in finding a better solution. A local solver might sometimes fail to find a solution and then it is not always clear whether that is caused by the problem being infeasible or by the solver failing to find a solution for a feasible problem. The presolve algorithm may reveal inconsistent constraints and/or variable bounds and hence identify a problem as infeasible.
The nonlinear presolve algorithm is switched off by default. To use it you should switch on the AIMMS general solvers option Nonlinear Presolve .
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