scholarly journals Parallelizing Subgradient Methods for the Lagrangian Dual in Stochastic Mixed-Integer Programming

2021 ◽  
Vol 3 (1) ◽  
pp. 1-22
Author(s):  
Cong Han Lim ◽  
Jeffrey T. Linderoth ◽  
James R. Luedtke ◽  
Stephen J. Wright

The dual decomposition of stochastic mixed-integer programs can be solved by the projected subgradient algorithm. We show how to make this algorithm more amenable to parallelization in a master-worker model by describing two approaches, which can be combined in a natural way. The first approach partitions the scenarios into batches and makes separate use of subgradient information for each batch. The second approach drops the requirement that evaluation of function and subgradient information is synchronized across the scenarios. We provide convergence analysis of both methods. We also evaluate their performance on two families of problems from SIPLIB on a single server with 32 single-core worker processes, demonstrating that when the number of workers is high relative to the number of scenarios, these two approaches (and their synthesis) can significantly reduce running time.

Author(s):  
Milan Hladík

Traditionally, game theory problems were considered for exact data, and the decisions were based on known payoffs. However, this assumption is rarely true in practice. Uncertainty in measurements and imprecise information must be taken into account. The interval-based approach for handling such uncertainties assumes that one has lower and upper bounds on payoffs. In this paper, interval bimatrix games are studied. Especially, we focus on three kinds of support set invariancy. Support of a mixed strategy consists of that pure strategies having positive probabilities. Given an interval-valued bimatrix game and supports for both players, the question states as follows: Does every bimatrix game instance have an equilibrium with the prescribed support? The other two kinds of invariancies are slight modifications: Has every bimatrix game instance an equilibrium being a subset/superset of the prescribed support? It is computationally difficult to answer these questions: the first case costs solving a large number of linear programs or mixed integer programs. For the remaining two cases a sufficient condition and a necessary condition are proposed, respectively.


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