Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequalities

Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequalities

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Article ID: iaor2013263
Volume: 7
Issue: 1
Start Page Number: 185
End Page Number: 206
Publication Date: Jan 2013
Journal: Optimization Letters
Authors: ,
Keywords: programming (minimax), Ky Fan inequality
Abstract:

In this paper we present the relaxed inertial proximal algorithm for Ky Fan minimax inequalities. Based on Opial lemma, we propose a weak convergence result to a solution of the problem by eliminating in the algorithm (RIPAFAN) the Browder–Halpern’s factor of contraction. We present after, a first result of strong convergence by adding a strong monotonicity condition. Secondly, we eliminate the strong monotonicity and add a Browder–Halpern’s contraction factor in the algorithm (RIPAFAN) and then ensure the strong convergence to a selected solution with respect to the contraction factor. Some examples are proposed. The first one concerns the convex minimization where the objective function is only controlled with a provided well conditioning. In the second one, we propose monotone set‐valued variational inequalities. The last example deals with the problem of fixed point for a nonexpansive set‐valued operator.

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