Resolution of Constrained Non-linear Optimization Problems Using Direct Search Methods Combined with New Measures to Admissibility in Filters Method

Abstract

Constrained non-linear optimization problems frequently appears in several areas, from engineering to economics, among others. These problems are usually solved using penalty or barrier methods which construct a sequence of unconstrained problems, which are then solved using unconstrained optimization methods. These methods attempts to minimize the objective function and the constraint violation functions simultaneously, defining a sequence of new objective function which includes information about the objective function and these violations. More recently filters method are used to solve constrained non-linear optimization problems. Filters method, introduced by Fletcher and Leyffer in 2002, have been widely used in several areas of constrained non-linear optimization. These methods deals with this kind of optimization problem as bi-objective, with two functions to minimize: the objective function and a continuous function that aggregates the constraint violation functions. Considering the possibility of information about the derivatives being not accessible, because the objective function and/or the constraints functions of being non smooth, non continuous, non convex and/or with many local minimums then direct search methods must be used in internal processes, in a method as much as in the other. Audet and Dennis in 2004 have presented the first filters method for direct search nonlinear programming, based on pattern search methods. Motivated by this work we have developed a new direct search method, based on simplex methods, for general constrained optimization, that combines the features of the simplex method and filters method, [1] and [2]. This work presents a new variant of these methods which combines the filters method with other direct search methods and are proposed some alternatives to aggregate the constraint violation functions.