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Stochastic quasi-subgradient method for stochastic quasi-convex feasibility problems

  • * Corresponding author: Yaohua Hu

    * Corresponding author: Yaohua Hu

The first author is supported in part by the Zhejiang Provincial Natural Science Foundation of China (LY18A010030), Scientific Research Fund of Zhejiang Provincial Education Department (19060042-F) and Science Foundation of Zhejiang Sci-Tech University (19062150-Y).
The second author is supported in part by the Foundation for High-level Talents of Chongqing University of Art and Sciences (P2017SC01), Chongqing Key Laboratory of Group and Graph Theories and Applications, and Key Laboratory of Complex Data Analysis and Artificial Intelligence of Chongqing Municipal Science and Technology Commission.
The third author is supported in part by the National Natural Science Foundation of China (12071306, 11871347), Natural Science Foundation of Guangdong Province of China (2019A1515011917, 2020B1515310008, 2020A1515010372), Project of Educational Commission of Guangdong Province of China (2019KZDZX1007), Natural Science Foundation of Shenzhen (JCYJ20190808173603590) and Interdisciplinary Innovation Team of Shenzhen University.

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  • The feasibility problem is at the core of the modeling of many problems in various disciplines of mathematics and physical sciences, and the quasi-convex function is widely applied in many fields such as economics, finance, and management science. In this paper, we consider the stochastic quasi-convex feasibility problem (SQFP), which is to find a common point of infinitely many sublevel sets of quasi-convex functions. Inspired by the idea of a stochastic index scheme, we propose a stochastic quasi-subgradient method to solve the SQFP, in which the quasi-subgradients of a random (and finite) index set of component quasi-convex functions at the current iterate are used to construct the descent direction at each iteration. Moreover, we introduce a notion of Hölder-type error bound property relative to the random control sequence for the SQFP, and use it to establish the global convergence theorem and convergence rate theory of the stochastic quasi-subgradient method. It is revealed in this paper that the stochastic quasi-subgradient method enjoys both advantages of low computational cost requirement and fast convergence feature.

    Mathematics Subject Classification: Primary: 65K05, 90C26; Secondary: 49M37.

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