We address the nonconvex optimization problem with the cost function and equality and inequality constraints given by d.c. functions. The linear space of d.c. functions possesses a number of very attractive properties. For example, every continuous function can be approximated at any desirable accuracy by a d.c. function and any twice differentiable function belongs to the DC space. In addition, any lower semicontinuous (l.s.c.) function can be approximated at any precision by a sequence of continuous functions. Furthermore, provided that for the optimization problem under study we proposed the new Global Optimality Conditions (GOCs), which have been published in the English and Russian languages. The natural question arises: is it possible to construct a computational scheme based on the GOCs (otherwise, what are they for?) that would allow us not only to generate critical points (like the KKT-vectors) but to escape any local pitfall, which makes it possible to reach a global solution to the problem in question? First of all, we recall that with the help of the Theory of Exact Penalization, the original d.c. problem was reduced to a problem without constraints. Moreover, it can be readily seen that this penalized problem is a d.c. problem as well. Furthermore, special Local Search Methods (LSMs) were developed and substantiated in view of their convergence features. In addition, the GOCs were generalized for the minimizing sequences in the penalized problem. A special theoretical method was proposed and its convergence properties were studied. We developed a Global Search Scheme (GSS) based on all theoretical results presented above, and, moreover, we were lucky to prove that the sequence produced by the GSS turned out to be minimizing in the original d.c. optimization problem. Finally, we developed a Global Search Method (GSM), combining the special LSM and the GSS proposed. The convergence of the GSM is also investigated under some natural assumptions. The first results of numerical testing of the approach will be demonstrated.
展开▼
