On Solving the Convex Semi-Infinite Minimax Problems via Superlinear 𝒱𝒰 Incremental Bundle Technique with Partial Inexact Oracle

Ming Huang (), Jinlong Yuan (), Sida Lin, Xijun Liang () and Chongyang Liu
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Ming Huang: School of Science, Dalian Maritime University, Dalian 116026, P. R. China
Jinlong Yuan: School of Science, Dalian Maritime University, Dalian 116026, P. R. China
Sida Lin: School of Science, Dalian Maritime University, Dalian 116026, P. R. China2School of Control Science and Engineering, Dalian University of Technology, Dalian 116024, P. R. China
Xijun Liang: College of Science, China University of Petroleum, Qingdao 266580, P. R. China
Chongyang Liu: School of Mathematics and Information Science, Shandong Technology and Business University, Yantai 264005, P. R. China

Asia-Pacific Journal of Operational Research (APJOR), 2021, vol. 38, issue 05, 1-32

Abstract: In this paper, we study convex semi-infinite programming involving minimax problems. One of the difficulties in solving these problems is that the maximum type functions are not differentiable. Due to the nonsmooth nature of the problem, we apply the special proximal bundle scheme on the basis of 𝒱𝒰-decomposition theory to solve the nonsmooth convex semi-infinite minimax problems. The proposed scheme requires an evaluation within some accuracy for all the components of the objective function. Regarding the incremental method, we only need one component function value and one subgradient which are estimated to update the bundle information and produce the search direction. Under some mild assumptions, we present global convergence and local superlinear convergence of the proposed bundle method. Numerical results of several example problems are reported to show the effectiveness of the new scheme.

Keywords: Convex optimization; nonsmooth optimization; 𝒱𝒰-decomposition; semi-infinite minimax programming; bundle method; superlinear convergence (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0217595921400157

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