We present a method for computing lower bounds in the progressive hedging algorithm (PHA) for two-stage and multi-stage stochastic mixed-integer programs. Computing lower bounds in the PHA allows one to assess the quality of the solutions generated by the algorithm contemporaneously. The lower bounds can be computed in any iteration of the algorithm by using dual prices that are calculated during execution of the standard PHA. We report computational results on stochastic unit commitment and stochastic server location problem instances, and explore the relationship between key PHA parameters and the quality of the resulting lower bounds. Abstract We present a method for computing lower bounds in the Progressive Hedging Algorithm (PHA) for two-stage and multi-stage stochastic mixedinteger programs. Computing lower bounds in the PHA allows one to assess the quality of the solutions generated by the algorithm contemporaneously. The lower bounds can be computed in any iteration of the algorithm by using dual prices that are calculated during execution of the standard PHA. We report computational results on stochastic unit commitment and stochastic server location problem instances, and explore the relationship between key PHA parameters and the quality of the resulting lower bounds.

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In this second portion of a two-part analysis of a scalable computational approach to stochastic unit commitment (SUC), we focus on solving stochastic mixedinteger programs in tractable run-times. Our solution technique is based on Rockafellar and Wets' progressive hedging algorithm, a scenario-based decomposition strategy for B Jean-Paul Watson 123 K. Cheung et al.solving stochastic programs. To achieve high-quality solutions in tractable run-times, we describe critical, novel customizations of the progressive hedging algorithm for SUC. Using a variant of the WECC-240 test case with 85 thermal generation units, we demonstrate the ability of our approach to solve realistic, moderate-scale SUC problems with reasonable numbers of scenarios in no more than 15 min of wall clock time on commodity compute platforms. Further, we demonstrate that the resulting solutions are high-quality, with costs typically within 1-2.5 % of optimal. For larger test cases with 170 and 340 thermal generators, we are able to obtain solutions of similar quality in no more than 25 min of wall clock time. A major component of our contribution is the public release of the optimization model, associated test cases, and algorithm results, in order to establish a rigorous baseline for both solution quality and run times of SUC solvers.

Abstract. In this paper, we study deterministic dynamic lot-sizing problems with service level constraints on the total number of periods in which backorders can occur over the finite planning horizon. We give a natural mixed integer programming formulation for the single item problem (LS-SL-I) and study the structure of its solution. We show that an optimal solution to this problem can be found in O(n 2 κ) time, where n is the planning horizon and κ = O(n) is the maximum number of periods in which demand can be backordered. Using the proposed shortest path algorithms, we develop alternative tight extended formulations for LS-SL-I and one of its relaxations, which we refer to as uncapacitated lot sizing with setups for stocks and backlogs. We show that this relaxation also appears as a substructure in a lot-sizing problem which limits the total amount of a period's demand met from a later period, across all periods. We report computational results that compare the natural and extended formulations on multi-item service-level constrained instances.

The capacitated-facilities location problem (CFLP) deals with opening facilities with a finite capacity to serve a set of customers. This paper addresses the discrete CFLP when the opened facilities are unreliable, i.e. they are unavailable to provide service to customers. Such problems have gained prominence in the recent past owing to their application in the area of supply-chain disruptions. A stochastic programming formulation for the CFLP with unreliable facilities is presented and the benefit of investing in redundant facility locations is demonstrated. A sampling-based algorithm called the sample average approximation algorithm is used to approximately solve this model and present computational results.