Advanced Methods for Nonlinear Programs

Problems of nonlinear programs

For non-convex nonlinear mathematical programs (NLPs), nonlinear solvers have no guarantee of returning the global optimum. Due to the local search algorithms employed by nonlinear solvers, their solution process depends on the starting point provided by the user. Nonlinear solvers can, therefore, easily end up in, non-unique, local optima, or, even worse, may not even find a feasible solution for a given starting point.

How to counteract?

To counteract these facts, a number of possible actions can be taken.

• Use a global solver, such as BARON, to solve the NLP. Global solvers, however, usually only work well on relatively small NLP problems.

• Use a multistart algorithm to solve the NLP problem for multiple starting points in order to have a better chance to find the global optimum.

• Use the AIMMS Presolver to reduce the problem size and tighten the bounds of the remaining variables and constraints of the NLP. This will reduce the space which the nonlinear solver needs to search in order to find an optimal solution.

This chapter

This chapter discusses the presolve techniques for nonlinear programs available in AIMMS. The chapter also discusses the multistart algorithm built into AIMMS. Using the multistart algorithm will increase the total solution time, but, in general, will also improve the solution found by nonlinear solvers.

• The AIMMS Presolver
• The AIMMS Multistart Algorithm
• Control Parameters That Influence the Multistart Algorithm