Derivative-free optimization and filter methods to solve nonlinear constrained problems

摘要

In real optimization problems, usually the analytical expression of the objective function is not known, norits derivatives, or they are complex. In these cases it becomes essential to use optimization methods wherethe calculation of the derivatives, or the verification of their existence, is not necessary: the Direct SearchMethods or Derivative-free Methods are one solution.When the problem has constraints, penalty functions are often used. Unfortunately the choice of thepenalty parameters is, frequently, very difficult, because most strategies for choosing it are heuristicsstrategies. As an alternative to penalty function appeared the filter methods. A filter algorithm introducesa function that aggregates the constrained violations and constructs a biobjective problem. In this problemthe step is accepted if it either reduces the objective function or the constrained violation. This implies thatthe filter methods are less parameter dependent than a penalty function.In this work, we present a new direct search method, based on simplex methods, for general constrainedoptimization that combines the features of the simplex method and filter methods. This method does notcompute or approximate any derivatives, penalty constants or Lagrange multipliers. The basic idea ofsimplex filter algorithm is to construct an initial simplex and use the simplex to drive the search. Weillustrate the behavior of our algorithm through some examples. The proposed methods were implementedin Java.