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Interior proximal methods for quasiconvex optimization. (English) Zbl 1279.90135
A generalized proximal point algorithm (PPA) for the minimization of a nonconvex function on a feasible set is investigated. It is interesting to know if methods also converge under essentially weaker assumptions. The present paper deals with the convergence analysis of both PPA and its extension, to optimization problems with only quasiconvex objectives. This includes several results on well-definedness; that is, solvability of the subproblems as well as their uniqueness and existence of solutions in int\((K)\) for the zone coercive case. The authors illustrate that in some sense quasi-convexity is the weakest assumption that permits to obtain results presented in this paper.
Apart from numerical advantage for nonconvex problems, the fact that well-definedness and convergence can principally be obtained under weaker conditions than those in literature seems to be interesting, see also [M. Fukushima and H. Mine, Int. J. Syst. Sci. 12, 989–1000 (1981; Zbl 0467.65028)] and [A. Kaplan and R. Tichatschke, J. Glob. Optim. 13, No. 4, 389–406 (1998; Zbl 0916.90224)].

MSC:
90C26 Nonconvex programming, global optimization
90C51 Interior-point methods
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