A Linear Decision-Based Approximation Approach to Stochastic Programming

Abstract

Stochastic optimization, especially multistage models, is well known to be computationally excruciating. Moreover, such models require exact specifications of the probability distributions of the underlying uncertainties, which are often unavailable. In this paper, we propose tractable methods of addressing a general class of multistage stochastic optimization problems, which assume only limited information of the distributions of the underlying uncertainties, such as known mean, support, and covariance. One basic idea of our methods is to approximate the recourse decisions via decision rules. We first examine linear decision rules in detail and show that even for problems with complete recourse, linear decision rules can be inadequate and even lead to infeasible instances. Hence, we propose several new decision rules that improve upon linear decision rules, while keeping the approximate models computationally tractable. Specifically, our approximate models are in the forms of the so-called second-order cone (SOC) programs, which could be solved efficiently both in theory and in practice. We also present computational evidence indicating that our approach is a viable alternative, and possibly advantageous, to existing stochastic optimization solution techniques in solving a two-stage stochastic optimization problem with complete recourse.Subject classifications: programming: stochastic. Area of review: Stochastic Models.

Robust optimality of linear saturated control in uncertain linear network flows

“…The conservative approach of Section V reminds the Soyster decomposition [12], used in robust linear programming. Also, the notion of feedback in control, present in this work, reminds the notion of recourse used in robust optimization [7]. Concerning the nature of the problem, we wish to emphasize our reversed perspective: given a solution (the saturated control policy) we ask for the objective function that makes the solution optimal.…”

“…Let us now explain more in details the notion of optimality of a saturated control mentioned in Problem 1. Let U and W be the sets of measurable controls and demands as in the Introduction (after equation (3)), and let Γ be the set of nonanticipative strategies for the player 1 (see (7), replacing B by W , A by U , and β by u). The (lower) value function for the differential game is then…”

Robust Storage Assignment in Unit-Load Warehouses

“…Chen et al (2008) show that a feasible stochastic optimization problem can become infeasible under a linear decision rule. Even if problem (3) is feasible, it is not clear whether there exists a linear decision rule that is feasible.…”

“…This is justified by the significant growth in this area of research (see, for instance, Soyster 1973;Ben-Tal and Nemirovski 1998, 2000Sim 2003, 2004;Chen et al 2008;Chen and Sim 2009;Goh and Sim 2010;El-Ghaoui and Lebret 1997;El-Ghaoui et al 1998;Erdogan and Iyengar 2006). Robust optimization has also been implemented in a dynamic setting that involves decision making in stages.…”

Optimization Under Probabilistic Envelope Constraints

“…This is in contrast to the post-analysis approach (e.g., Paschalidis and Kang 2005), where one obtains a solution using alternative methods such as robust optimization or standard chance constraints, and then analyzes the tail probability of the constraint violation for the obtained solution. Recently, there has been some work on robust optimization that takes into account the probabilistic requirements on the solution, e.g., (Chen et al 2007(Chen et al , 2008Bertsimas and Brown 2009). These papers construct uncertainty sets such that the obtained solution is guaranteed to satisfy certain probabilistic requirements.…”