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inexact stochastic mirror descent for two-stage nonlinear stochastic programs

MATHEMATICAL PROGRAMMING 2021年第1-2期187卷 533-577页

作者： Guigues, Vincent FGV Sch Appl Math Rio De Janeiro Brazil

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.

关键词： inexact cuts for value functions inexact stochastic mirror descent Strong concavity of the dual function Stochastic programming

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Stochastic Dynamic Cutting Plane for Multistage Stochastic Convex Programs

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2021年第2期189卷 513-559页

作者： Guigues, Vincent Monteiro, Renato D. C. FGV Sch Appl Math Praia De Botafogo RJ Brazil Georgia Inst Technol Atlanta GA 30332 USA

We introduce Stochastic Dynamic Cutting Plane (StoDCuP), an extension of the Stochastic Dual Dynamic Programming (SDDP) algorithm to solve multistage stochastic convex optimization problems. At each iteration, the algorithm builds lower bounding affine functions not only for the cost-to-go functions, as SDDP does, but also for some or all nonlinear cost and constraint functions. We show the almost sure convergence of StoDCuP. We also introduce an inexact variant of StoDCuP where all subproblems are solved approximately (with bounded errors) and show the almost sure convergence of this variant for vanishing errors. Finally, numerical experiments are presented on nondifferentiable multistage stochastic programs where inexact StoDCuP computes a good approximate policy quicker than StoDCuP while SDDP and the previous inexact variant of SDDP combined with Mosek library to solve subproblems were not able to solve the differentiable reformulation of the problem.

关键词： Stochastic programming inexact cuts for value functions SDDP inexact SDDP

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inexact cuts IN STOCHASTIC DUAL DYNAMIC PROGRAMMING

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SIAM JOURNAL ON OPTIMIZATION 2020年第1期30卷 407-438页

作者： Guigues, Vincent FGV Appl Math BR-22250900 Rio De Janeiro Brazil

We introduce an extension of stochastic dual dynamic programming (SDDP) to solve stochastic convex dynamic programming equations. This extension applies when some or all primal and dual subproblems to be solved along the forward and backward passes of the method are solved with bounded errors (inexactly). This inexact variant of SDDP is described for both linear problems (the corresponding variant being denoted by ISDDP-LP) and nonlinear problems (the corresponding variant being denoted by ISDDP-NLP). We prove convergence theorems for ISDDP-LP and ISDDP-NLP for both bounded and asymptotically vanishing errors. Finally, we present the results of numerical experiments comparing SDDP and ISDDP-LP on a portfolio problem with direct transaction costs modeled as a multistage stochastic linear optimization problem. In these experiments, ISDDP-LP allows us to strike a different balance between policy quality and computing time, trading off the former for the latter.

关键词： stochastic programming inexact cuts for value functions bounding epsilon-optimal dual solutions SDDP inexact SDDP

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inexact cuts IN STOCHASTIC DUAL DYNAMIC PROGRAMMING APPLIED TO MULTISTAGE STOCHASTIC NONDIFFERENTIABLE PROBLEMS

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SIAM JOURNAL ON OPTIMIZATION 2021年第3期31卷 2084-2110页

作者： Guigues, Vincent Monteiro, Renato Svaiter, Benar Fundacao Getulio Vargas Sch Appl Math 190 Praia Botafogo Rio De Janeiro Brazil Georgia Inst Technol Sch Ind & Syst Engn Atlanta GA 30332 USA IMPA Estr Dona Castorina 110 Rio De Janeiro Brazil

stochastic dual dynamic programming (SDDP) called ISDDP was introduced which uses approximate (instead of exact with SDDP) primal dual solutions of the problems solved in the forward and backward passes of the method. That variant of SDDP was studied in [V. Guigues, SIAM J. Optim., 30 (2020), pp. 407--438] for linear and for differentiable nonlinear multistage stochastic programs (MSPs). In this paper, we extend ISDDP to nondifferentiable MSPs. We first provide formulas for inexact cuts for value functions of convex nondifferentiable optimization problems. We then combine these cuts with SDDP to describe ISDDP for nondifferentiable MSPs and analyze the convergence of the method. More precisely, for a problem with T stages, we show that for errors bounded from above by \varepsilon , the limit superior and limit inferior of sequences of upper and lower bounds on the optimal value of the problem are at most at distance 3\varepsilon T to the optimal value and that for asymptotically vanishing errors ISDDP converges to an optimal policy. Finally, we present the results of encouraging numerical experiments on a multistage nondifferentiable stochastic convex program solved using exact SDDP and the proposed ISDDP.

关键词： stochastic optimization SDDP inexact cuts for value functions inexact SDDP

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