Persistence in discrete optimization under data uncertainty

  • Authors:

  • Dimitris Bertsimas;Karthik Natarajan;Chung-Piaw Teo

  • Affiliations:

  • (Boeing Professor of Operations Research) Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA;Department of Mathematics, National University of Singapore, 02139, Cambridge, MA, Singapore;Department of Decision Sciences, Business School, National University of Singapore, 117543, Cambridge, MA, Singapore

  • Venue:
  • Mathematical Programming: Series A and B
  • Year:
  • 2006

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Abstract

An important question in discrete optimization under uncertainty is to understand the persistency of a decision variable, i.e., the probability that it is part of an optimal solution. For instance, in project management, when the task activity times are random, the challenge is to determine a set of critical activities that will potentially lie on the longest path. In the spanning tree and shortest path network problems, when the arc lengths are random, the challenge is to pre-process the network and determine a smaller set of arcs that will most probably be a part of the optimal solution under different realizations of the arc lengths. Building on a characterization of moment cones for single variate problems, and its associated semidefinite constraint representation, we develop a limited marginal moment model to compute the persistency of a decision variable. Under this model, we show that finding the persistency is tractable for zero-one optimization problems with a polynomial sized representation of the convex hull of the feasible region. Through extensive experiments, we show that the persistency computed under the limited marginal moment model is often close to the simulated persistency value under various distributions that satisfy the prescribed marginal moments and are generated independently.