Multiplier Stabilization Applied to Two-Stage Stochastic Programs

Clara Lage (), Claudia Sagastizábal () and Mikhail Solodov ()
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Clara Lage: ENGIE E&P International [La Défense], CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, UP1 - Université Paris 1 Panthéon-Sorbonne
Claudia Sagastizábal: IMECC - Instituto de Matemática, Estatística e Computação Científica [Brésil] - UNICAMP - Universidade Estadual de Campinas = University of Campinas
Mikhail Solodov: IMPA - Instituto Nacional de Matemática Pura e Aplicada

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Abstract: In many mathematical optimization applications dual variables are an important output of the solving process, due to their role as price signals. When dual solutions are not unique, different solvers or different computers, even different runs in the same computer if the problem is stochastic, often end up with different optimal multipliers. From the perspective of a decision maker, this variability makes the prices signals less reliable and, hence, less useful. We address this issue for a particular family of linear and quadratic programs by proposing a solution procedure that, among all possible optimal multipliers, systematically yields the one with the smallest norm. The approach, based on penalization techniques of nonlinear programming, amounts to a regularization in the dual of the original problem. As the penalty parameter tends to zero, convergence of the primal sequence and, more critically, of the dual is shown under natural assumptions. The methodology is illustrated on a battery of two-stage stochastic linear programs.

Keywords: Multiplier stability; Dual regularization; Penalty method; Stochastic programming; Two-stage stochastic programming; Empirical approximations; Stabilité multiplicateur; Régularisation double; Méthode de pénalité; Programmation stochastique; Programmation stochastique en deux étapes; Approximations empiriques (search for similar items in EconPapers)
Date: 2020-06
Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-02900862v1
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Published in 2020

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