On convergence of a global search strategy for reverse convex problems

Nowadays specialists on optimization observe the persistent demands from the world of applications to create an effective apparatus for finding just a global solution to nonconvex problems in which there may exist local solutions located very far from a global one even up to the values of goal function. As well-known, the conspicuous limitation of convex optimization methods applied to nonconvex problems is their ability of being trapped at a local extremum or even a critical point depending on a starting point. In other words, the classical apparatus shows itself inoperative for new problems arising from practice. That is why, the development of nonconvex optimization took the way of tools initially generated in discrete optimization (DO) and for discrete optimization. So, DO gave some apparatus for continuous optimization (CO). Gradually, Branches & Bounds, cuts, and so forth ideas became very popular in nonconvex area of CO, although in some cases its turned out to be too much sensitive, for instance, with respect to the changing the size of a problem. In [8, 9, 10, 11, 12] it was proposed other approach to d.c. programming problems based on global optimality conditions (GOC) that has proved its effectiveness for numerous continuous (even dynamical) and discrete optimization problems [11, 13, 14, 15]. Nevertheless, the theoretical substantiation of the approach stays in some cases uncompleted. This concerns, in particular, convergence proofs for global search strategies based on GOC. This paper aims especially to full the lacuna for RCP. On the other hand, we demonstrate below the importance of new notion in optimization—the notion of resolving set, or a “good” approximation of the level surface of a convex function g(·). The crucial influence of the approximation on results of global search was transparently shown in [11, 12, 14, 15] by computational experiments. Here we would like