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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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##### References:
 [1] Auslender A., Haddou M.: An interior-proximal method for convex linearly constrained problems and its extension to variational inequalities. Math. Prog. 71(1), 77–100 (1995) · Zbl 0855.90095 [2] Bregman L.M.: The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7, 200–217 (1967) · Zbl 0186.23807 · doi:10.1016/0041-5553(67)90040-7 [3] Burachik R.S., Iusem A.N.: A generalized proximal point algorithm for the variational inequality problem in Hilbert space. SIAM J. Optim. 8(1), 197–216 (1998) · Zbl 0911.90273 · doi:10.1137/S1052623495286302 [4] Censor Y., Iusem A.N., Zenios S.A.: An interior point method with Bregman functions for the variational inequality problem with paramonotone operators. Math. Prog. 81(3), 373–400 (1998) · Zbl 0919.90123 [5] Censor Y., Zenios S.A.: Proximal minimization algorithm with D-functions. J. Optim. Theory Appl. 73(3), 451–464 (1992) · Zbl 0794.90058 · doi:10.1007/BF00940051 [6] Chen G., Teboulle M.: Convergence analysis of a proximal-like minimization algorithm using Bregman functions. SIAM J. Optim. 3(3), 538–543 (1993) · Zbl 0808.90103 · doi:10.1137/0803026 [7] Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) · Zbl 0582.49001 [8] Eckstein J.: Nonlinear proximal point algorithms using Bregman functions, with application to convex programming. Math. Oper. Res. 18(1), 202–226 (1993) · Zbl 0807.47036 · doi:10.1287/moor.18.1.202 [9] Fukushima M., Mine H.: A generalized proximal point algorithm for certain non-convex minimization problems. Int. J. Syst. Sci 12(8), 989–1000 (1981) · Zbl 0467.65028 · doi:10.1080/00207728108963798 [10] Güler O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29(2), 403–419 (1991) · Zbl 0737.90047 · doi:10.1137/0329022 [11] Hager W., Zhang H.: Self-adaptive inexact proximal point methods. Comput. Optim. Appl. 39(2), 161–181 (2008) · Zbl 1147.90414 · doi:10.1007/s10589-007-9067-3 [12] Kaplan A., Tichatschke R.: Stable Methods for Ill-Posed Variational Problems–Prox-Regularization of Elliptic Variational Inequalities and Semi-Infinite Optimization Problems. Akademie Verlag, Berlin (1994) · Zbl 0804.49011 [13] Kaplan A., Tichatschke R.: Proximal point methods and nonconvex optimization. J. Global Optim. 13(4), 389–406 (1998) · Zbl 0916.90224 · doi:10.1023/A:1008321423879 [14] Kaplan A., Tichatschke R.: Proximal point method and elliptic regularization. Nonlinear Anal. Theory Methods Appl. 71(10), 4525–4543 (2009) · Zbl 1214.47062 · doi:10.1016/j.na.2009.03.010 [15] Kaplan A., Tichatschke R.: Note on the paper: interior proximal method for variational inequalities on non-polyhedral sets. Discuss. Math. Diff. Inclusions Control Optim. 30, 51–59 (2010) · Zbl 1214.47063 · doi:10.7151/dmdico.1111 [16] Langenberg N.: On the cutting plane property and the Bregman proximal point algorithm. J. Convex Anal. 18(3), 601–619 (2011) · Zbl 1267.47096 [17] Langenberg N.: Pseudomonotone operators and the Bregman proximal point algorithm. J. Global Optim. 47(4), 537–555 (2010) · Zbl 1228.90081 · doi:10.1007/s10898-009-9470-7 [18] Langenberg N.: Convergence analysis of an extended auxiliary problem principle with various stopping criteria. Optim. Methods Softw. 26(1), 127–154 (2011) · Zbl 1238.65058 · doi:10.1080/10556780903406589 [19] Martinet B.: Régularisation d’inéquations variationelles par approximations successives. Revue française d’informatique et de recherche opérationnelle, série rouge 4(3), 154–158 (1970) [20] Rockafellar R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976) · Zbl 0358.90053 · doi:10.1137/0314056 [21] Solodov M.V., Svaiter B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Prog. 87(1), 189–202 (2000) · Zbl 0971.90062
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