Inexact stochastic mirror descent for two-stage nonlinear stochastic programs

作     者：Guigues, Vincent 

作者机构：FGV Sch Appl Math Rio De Janeiro Brazil 

出 版 物：《MATHEMATICAL PROGRAMMING》 (数学规划)

年 卷 期：2021年第187卷第1-2期

页      面：533-577页

核心收录：

学科分类：1201[管理学-管理科学与工程(可授管理学、工学学位)] 07[理学] 070104[理学-应用数学] 0835[工学-软件工程] 0701[理学-数学] 

基　　金：FGV Grant CNPq [311289/2016-9] FAPERJ [E-26/201.599/2014] 

主　　题：Inexact cuts for value functions Inexact stochastic mirror descent Strong concavity of the dual function Stochastic programming 

摘      要：We introduce an inexact variant of stochastic mirror descent (SMD), called inexact stochastic mirror descent (ISMD), to solve nonlinear two-stage stochastic programs where the second stage problem has linear and nonlinear coupling constraints and a nonlinear objective function which depends on both first and second stage decisions. Given a candidate first stage solution and a realization of the second stage random vector, each iteration of ISMD combines a stochastic subgradient descent using a prox-mapping with the computation of approximate (instead of exact for SMD) primal and dual second stage solutions. We provide two convergence analysis of ISMD, under two sets of assumptions. The first convergence analysis is based on the formulas for inexact cuts of value functions of convex optimization problems shown recently in Guigues (SIAM J. Optim. 30(1), 407-438, 2020). The second convergence analysis provides a convergence rate (the same as SMD) and relies on new formulas that we derive for inexact cuts of value functions of convex optimization problems assuming that the dual function of the second stage problem for all fixed first stage solution and realization of the second stage random vector, is strongly concave. We show that this assumption of strong concavity is satisfied for some classes of problems and present the results of numerical experiments on two simple two-stage problems which show that solving approximately the second stage problem for the first iterations of ISMD can help us obtain a good approximate first stage solution quicker than with SMD.