Mixed 0-1 Linear Programs Under Objective Uncertainty: A Completely Positive Representation
Operations Research, Vol. 59, No. 3, pp. 713-728, 2011
36 Pages Posted: 25 May 2016 Last revised: 13 Feb 2018
Date Written: August 7, 2009
In this paper, we analyze mixed 0-1 linear programs under objective uncertainty. The mean vector and the second moment matrix of the nonnegative objective coefficients is assumed to be known, but the exact form of the distribution is unknown. Our main result shows that computing a tight upper bound on the expected value of a mixed 0-1 linear program in maximization form with random objective is a completely positive program. This naturally leads to semidefinite programming relaxations that are solvable in polynomial time but provide weaker bounds. The result can be extended to deal with uncertainty in the moments and more complicated objective functions. Examples from order statistics and project networks highlight the applications of the model. Our belief is that the model will open an interesting direction for future research in discrete and linear optimization under uncertainty.
Keywords: Mixed 0-1 Linear Program; Moments; Completely Positive Program
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