Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization

Abstract

Abstract. This paper studies the general vector optimization problem of finding weakly efficient points for mappings in a Banach space Y, with respect to the partial order induced by a closed, convex, and pointed cone C C Y with nonempty interior. In order to find a solution of this problem, we introduce an auxiliary variational inequality problem for monotone, 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 by the combination of extragradient method for finding a solution to the variational inequality problem and 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, Lipschitz-continuous mapping, and finding weakly efficient points for suitable regularizations of the original mapping. We present both an absolute and a relative version in which the subproblems are solved only approximately. Weak convergence of the generated sequence to a weak efficient point is established under quite mild conditions. In addition, we also discuss an extension to Bregman-function-based hybrid approximate proximal algorithms for finding weakly efficient points for mappings.

An Inexact Spingarn’s Partial Inverse Method with Applications to Operator Splitting and Composite Optimization

“…The hybrid proximal extragradient (HPE) method of [3] has been used in the last few years as a framework for the analysis and development of several algorithms for solving monotone inclusion, saddle-point and convex optimization problems [3,4,5,6,7,8,9,10,11,12,13,14]. Next we present the HPE method.…”

“…The hybrid proximal extragradient (HPE) method of Solodov and Svaiter [3] is an inexact version of the Rockafellar's PPM which uses relative error tolerance criterion for solving each proximal subproblem instead of summable error condition. The HPE method has been used for many authors [3,4,5,6,7,8,9,10,11,12,13,14] as a framework for the design and analysis of several algorithms for monotone inclusion problems, variational inequalities, saddlepoint problems and convex optimization. Its iteration-complexity has been established recently by Monteiro and Svaiter [15] and, as a consequence, it has proved the iteration-complexity of various important algorithms in optimization (which use the HPE method as a framework) including Tseng's forward-backward method, Korpelevich extragradient method and the alternating direction method of multipliers (ADMM) [12,15,16].…”

Generalized Proximal Method for Efficient Solutions in Vector Optimization

“…Later, Ceng and Yao [6] generalized the results in [4] to the approximate ones as well as discussed an extension to Bregman function based proximal algorithms for solving a weak efficient solution of (1.3). Recently, the authors in [5] introduced and studied a certain hybrid approximate proximal method of finding weak efficient solutions to the convex constrained vector optimization problem by utilizing and developing iterative processes from the fixed-point theory for nonexpansive operators and combining them with algorithms for solving some classes of monotone variational inequalities. Another results in this direction are given in [9], where the authors developed new hybrid approximate proximal-type algorithms to find efficient (or Pareto) solutions to problems of the convex constrained vector optimization in both finitedimensional and infinite-dimensional Hilbert spaces.…”

Proximal point method for a special class of nonconvex multiobjective optimization functions

“…These types of problems have applications in the economy, industry, agriculture, and other fields; see [13]. [6] considered extensions of the proximal point method to the multiobjective setting, see also, [1,2,3,4,7,8,9,20] and references therein.…”