一般混合变分不等式的捆集近似算法

Abstract

Abstract:

In this paper, the authors consider a bundle proximal method for solving general mixed variational inequalities. The method is based on the auxiliary problem principle due to Cohen and the bundle Bregman proximal method for convex nonsmooth optimization due to Kiwiel. The strategy is to approximate, in the subproblems, the nonsmooth convex function f by a sequence of linear convex piecewise functions f_k, which is constructed from accumulated subgradient linearizations of f. As in the bundle Bregman proximal method for nonsmooth optimization, the method generates a sequence {x^k} by taking x^k to be an approximate minimizer of subproblems. This makes the subproblems more tractable. The authors first explain how to build a new iterative scheme and a stopping criterion to determine whether the current approximation is good enough. This criterion is different from that commonly used in the special case of nonsmooth optimization. The authors also prove that the convergence of the algorithm for the case that the mapping T satisfies the pseudo-Dunn property.

Key words: Iterative schemes, Proximal methods, Bundle methods, Strongly convex function, Pseudo-Dunn property

CLC Number:

90C25

XIA Fu-Quan, HUANG Na-Jing. A Bundle Proximal Method for Solving General Mixed Variational Inequalities[J].Acta mathematica scientia,Series A, 2011, 31(4): 866-879.

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References

[1] Aubin J P, Ekeland I. Applied Nonlinear Analysis. Now York: A Wiley Interscience Publication, 1984

[2] Alber Y I, Iusem A N, Solodov M V. On the projected subgradient method for nonsmooth convex optimization in a Hilbert space. Math Program, 1998, 81: 23--35

[3] Cohen G. Auxiliary problem principle extended to variational inequalities. Journal of Optimization Theory and Applications, 1988, 49: 325--333

[4] Cohen G. Nash equilibria: gradient and decomposition algorithms. Large Scale Systems, 1987, 12: 173--184

[5] Ding X P, Tan K K. A minimax inequality with application th existence of equilibrium point and fixed point theorems. Colloquium Math, 1992,  63: 233--247

[6] Ekeland I, Temam R. Convex Analysis and Variational Inequalities. Amsterdam: North-Holland, 1976

[7] Facchinri F, Pang J S. Finite Dimensional Variational Inequalities and Complementarity Problems. New York: Springer-Verlag, 2003

[8]} Gigola C, Gomez S. A regularization method for solving the finite convex min-max problem. SIAM Journal on Numerical Analysis, 1990, 27: 1621--1634

[9] Hartman P, Stampacchia G. On some nonlinear elliptic differential functional equations. Acta Mathematics, 1966, 115: 271--310

[10] Konnov I. A combined relaxation method for a class of nonlinear variational inequalities. Optimization, 2002, 51: 127--143

[11] Kiwiel K C. A bundle Bregman proximal method for convex nondifferentiable minimization. Math Program, 1999, 85: 241--258

[12] Kiwiel K C. Relaxation methods for strictly convex regularizations of piecewise linear programs. Appl Math Optim, 1998, 38: 239--259

[13] Kothe G. Topological Vector Spaces. Berlin, Germany: Springer-Verlag, 1983

[14] Panagiotopoulos P,  Stavroulakis G. New types of variational principles based on the notion of quasidifferentiablity. Acta Mech, 1994, 94: 171--194

[15] Rockafellar R T. Monotone operators and the proximal point algorith. SIAM J Control Optim, 1976, 14: 877--898

[16] Salmon G. Peturbed Auxiliary Problem Methods to Solve Generalized Variational Inequalities. Namur, Belgium: Presses                   Universitaires de Namur, 2001

[17] Salmon G, Strodiot J, Nguyen V H. A bundle method for solving variational inequalities. SIAM J Optim, 2004, 14(3): 869--893

[18] Solodov M V, Svaiter B F. A new projection method for variational inequality problems. SIAM J Control Optim, 1999, 37: 765--776

[19] Tseng P. Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J Control Optim, 1991, 29: 119--138

[20] Wang Y L, Xiu N H, Wang C Y.  Unified framework of extragradient type methods for pseudomonotone variational inequalities. Journal of Optimization Theory and Applications, 2001, 111: 641--656

[21] Xia F Q, Huang N J. Algorithm for solving a new class of general mixed variational inequalities in Banach spaces. Journal              of Computational and Applied Mathematics, 2008, 111: 632--642

[22] Xia F Q, Huang N J. A projected subgradient method for solving generalized mixed variational inequalities. Operations Research
Letters, 2008, 36: 637--642

[23] Zhu D L, Marcotte P. Cocoercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM J Optim, 1996, 6: 714--726

A Bundle Proximal Method for Solving General Mixed Variational Inequalities

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