A coarse solution of generalized semi-infinite optimization problems via robust analysis of marginal functions and global optimization

摘要

The aim of this work is to determine a coarse approximation to the optimal solution of a class of generalized semi-infinite optimization problems (GSIP) through a global optimization method by using fairly discontinuous penalty functions. Where the fairness of the discontinuities is characterized by the notions of robust analysis and standard measure theory. Generalized semi-infinite optimization problems have an infinite number of constraints, where the usually infinite index set of the constraints varies with respect to the problem variable; i.e. we have a set-valued map as and index set, in contrast to standard semi-infinite optimization (SIP) problems. These problems have very complex problem structures, at the same time, there are several classes of scientific, engineering, econimic, etc., problems which could be modelled in terms of (GSIP)s.Under general assumptions, the feasible set of a (GSIP) might not be closed nor connected. In fact, the feasible set is a closed set if the index map is lower semi-continuous. Several authors assume the lower semi-continuity of the index map for the derivation of numerical algorithms for (GSIP). However, in this work no exclusive assumption has been made to preserve the above nicer structures. Thus, the feasible set may not be closed and (GSIP) may not have a solution. However, one may be interested to determine a generalized minimizer or a minimizing sequence of GSIP. For this purpose, two penalty approaches have been proposed.In the first approach (mainly conceptual), there is defined a discontinuous penalty function based on the marginal function of a certain auxiliary parametric semi-infinite optimization problem (PSIP). In the second approach (based on discretization), we define two penalty functions: one based on the marginal function of the lower level problem and, a second, based on the feasible set of (GSIP). The relationships of these penalty problems with the (GSIP) have been investigated through minimizing sequences. In the two penalty approaches we need to deal with discontinuous optimization problems. The numerical treatment of these discontinuous optimization problems can be done by using the Integral Global Optimization Method (IGOM); in particular, through the software routine called BARLO (of Hichert). However, to use BARLO or IGOM we need to verify certain robustness properties of the objective functions of the penalty problems.Hence, one major contribution of this work is a study of robustness properties of marginal value functions and set-valued maps with given structures - extending the theory of robust analysis of Chew and Zheng. At the same time, an effort has been made to find out corresponding robustness results to some standard continuity notions of functions and set-valued maps.To show the viability of the proposed approach, numerical experiments are made using the penalty-discretization approach.