The Million-Variable “March” for Stochastic Combinatorial Optimization

Abstract

Combinatorial optimization problems have applications in a variety of sciences and engineering. In the presence of data uncertainty, these problems lead to stochastic combinatorial optimization problems which result in very large scale combinatorial optimization problems. In this paper, we report on the solution of some of the largest stochastic combinatorial optimization problem consisting of over a million binary variables. While the methodology is quite general, the specific application with which we conduct our experiments arises in stochastic server location problems. The main observation is that stochastic combinatorial optimization problems are comprised of loosely coupled subsystems. By taking advantage of the loosely coupled structure, we show that decomposition-coordination methods provide highly effective algorithms, and surpass the scalability of even the most efficiently implemented backtracking search algorithms.

An Approximate Dynamic Programming Algorithm for the Allocation of High-Voltage Transformer Spares in the Electric Grid

“…The experimental evidence in Louveaux & Peeters (1992), Laporte et al (1994), and Ntaimo & Sen (2005) shows this for integer stochastic programming based algorithms. Interestingly, the results in Ntaimo & Sen (2005) suggest that computational difficulties do not necessarily arise with a large sample space but with a high dimensional first-stage decision vector x 0 = (x 0a ) a∈A .…”

“…Sen (2005) provides a comprehensive overview of the state of the art in stochastic integer programming. Stochastic facility location problems (Louveaux & Peeters (1992), Laporte et al (1994), Ntaimo & Sen (2005)) are among the many applications of stochastic integer programming. In that context the problem is to open a number of facilities that can be used to satisfy random client demand.…”

“…For example, the largest instances solved in Ntaimo & Sen (2005) have 10 and 15 possible facility locations whereas we solve instances with up to 71 candidate locations. Our problem becomes much larger when we add dimensions such as transformer type (e.g.…”

“…self-monitoring maintenance). It is clear from the CPU times reported in Ntaimo & Sen (2005) that the number of possible facility locations has a decisive impact on run times. These findings are consistent with the results reported in (see also Topaloglu (2001)) which show that Benders decomposition becomes quite slow as the dimensionality of the second stage resource vector grows.…”

A comparative study of decomposition algorithms for stochastic combinatorial optimization

“…In contrast, for SCO problems in which the secondstage includes binary decision variables, new decomposition algorithms are necessary. In a recent paper, [13] have provided initial evidence that the disjunctive decomposition (D 2 ) algorithm [17] can provide better performance to direct methods for at least one class of SCO problems (stochastic server location problems or SSLPs).…”