Abstract
This paper studies a general vector optimization problem of finding weakly efficient points for mappings from Hilbert spaces to arbitrary Banach spaces, where the latter are partially ordered by some closed, convex, and pointed cones with nonempty interiors. To find solutions of this vector optimization problem, we introduce an auxiliary variational inequality problem for a monotone and Lipschitz continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem under consideration by combining an extragradient method to find a solution of the variational inequality problem and an approximate proximal point method for finding a root of a maximal monotone operator. In this hybrid approximate proximal method, the subproblems consist of finding approximate solutions to the variational inequality problem for monotone and Lipschitz continuous mapping, and then finding weakly efficient points for a suitable regularization of the original mapping. We present both absolute and relative versions of our hybrid algorithm in which the subproblems are solved only approximately. The weak convergence of the generated sequence to a weak efficient point is established under quite mild assumptions. In addition, we develop some extensions of our hybrid algorithms for vector optimization by using Bregman-type functions.
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Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15(4), 953–970 (2005)
Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51, 479–494 (2000)
Grana Drummond, L.M., Svaiter, B.F.: A steepest descent method for vector optimization. J. Comput. Appl. Math. 175, 395–414 (2005)
Grana Drummond, L.M., Iusem, A.N.: A projected gradient method for vector optimization problems. Comput. Optim. Appl. 28, 5–30 (2004)
Eckstein, J.: Approximate iterations in Bregman-function-based proximal algorithms. Math. Program. 83, 113–123 (1998)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Burachik, R.S., Iusem, A.N., Svaiter, B.F.: Enlargement of monotone operators with applications to variational inequalities. Set-Valued Anal. 5, 159–180 (1997)
Burachik, R.S., Scheimberg, S., Svaiter, B.F.: Robustness of the hybrid extragradient proximal point algorithm. J. Optim. Theory Appl. 111, 117–136 (2001)
He, B.S.: Inexact implicit methods for monotone general variational inequalities. Math. Program. 86, 199–217 (1999)
Jahn, J.: Theory of vector maximization: Various concepts of efficient solutions. In: Gal, T., Stewart, T.J., Hanne, T. (eds.): Multicriteria Decision Making. Advances in MCDM Models, Algorithms, Theory, and Applications, pp. 2-1–2-32. Kluwer, Boston (1999)
Solodov, M.V., Svaiter, B.F.: An inexact hybrid extragradient-proximal point algorithm using the enlargement of a maximal monotone operator. Set-Valued Anal. 7, 323–345 (1999)
Burachik, R.S., Iusem, A.N.: Set-Valued Mappings and Enlargements of Monotone Operators. Optimization and Its Applications, vol. 8. Springer, Berlin (2008)
Chen, G., Teboulle, M.: A proximal-based decomposition method for convex minimization problems. Math. Program. 64, 81–100 (1994)
Da Silva Silva, P.J., Eckstein, J., Humes, C.: Rescaling and stepsize selection in proximal methods using separable generalized distance. SIAM J. Optim. 12, 238–261 (2001)
Solodov, M.V., Svaiter, B.F.: A hybrid projection-proximal point algorithm. J. Convex Anal. 6, 59–70 (1999)
Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. 87, 189–202 (2000)
Solodov, M.V., Svaiter, B.F.: An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions. Math. Oper. Res. 25(2), 214–230 (2000)
Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)
Benker, H., Hamel, A., Tammer, C.: An algorithm for vectorial control approximation problems. In: Fandel, G., Gal, T. (eds.) Multiple Criteria Decision Making, pp. 3–12. Springer, Berlin (1997)
Kyono, M., Fukushima, M.: Nonlinear proximal decomposition method for convex programming. J. Optim. Theory Appl. 106, 357–372 (2000)
Butnariu, D., Iusem, A.N.: Totally Convex Functions for Fixed Point Computation and Infinite Dimensional Optimization. Kluwer, Dordrecht (1997)
Ceng, L.C., Yao, J.C.: Approximate proximal methods in vector optimization. Eur. J. Oper. Res. 183(1), 1–19 (2007)
Nadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz continuous monotone mappings. SIAM J. Optim. 16, 1230–1241 (2006)
Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Matecon 12, 747–756 (1976)
Burachik, R.S., Lopes, J.O., Svaiter, B.F.: An outer approximation method for the variational inequality problem. SIAM J. Control Optim. 43, 2071–2088 (2005)
Ceng, L.C., Cubiotti, P., Yao, J.C.: An implicit iterative scheme for monotone variational inequalities and fixed point problems. Nonlinear Anal. 69(8), 2445–2457 (2008)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Springer, Berlin (2006)
Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. (2009), published online
Bolintineanu, S.: Approximate efficiency and scalar stationarity in unbounded nonsmooth convex vector optimization problems. J. Optim. Theory Appl. 106, 265–296 (2000)
Bolintineanu, S.: Vector variational principles, ε-efficiency and scalar stationarity. J. Convex Anal. 8, 71–85 (2001)
Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Econom. and Math. Syst., vol. 319. Springer, Berlin (1989)
Schu, J.: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Austral. Math. Soc. 43, 153–159 (1991)
Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)
Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Math., vol. 1364. Springer, Berlin (1988)
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Communicated by F. Giannessi.
L.C. Ceng research was partially supported by the Leading Academic Discipline Project of Shanghai Normal University (DZL707), Innovation Program of Shanghai Municipal Education Commission Grant (09ZZ133), National Science Foundation of China (10771141), PhD Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), and Shanghai Leading Academic Discipline Project (S30405).
B.S. Mordukhovich research was partially supported by the USA National Science Foundation under Grant DMS-0603896.
J.C. Yao research was partially supported by Grant NSC 98-2923-E-110-003-MY3.
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Ceng, L.C., Mordukhovich, B.S. & Yao, J.C. Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization. J Optim Theory Appl 146, 267–303 (2010). https://doi.org/10.1007/s10957-010-9667-4
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DOI: https://doi.org/10.1007/s10957-010-9667-4