Inexact Stochastic Mirror Descent for two-stage nonlinear stochastic programs

Abstract

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 propose two variants of ISMD and show the convergence of these variants to the optimal value of the stochastic program. We show in particular that under some assumptions, ISMD has the same convergence rate as SMD. The first variant of ISMD and its convergence analysis are based on the formulas for inexact cuts of value functions of convex optimization problems shown recently in [4]. The second variant of ISMD and the corresponding convergence analysis rely 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. [4] V. Guigues, Inexact decomposition methods for solving deterministic and stochastic convex dynamic programming equations, arXiv, available at arXiv:1707.00812, 2017.