Mixed 0-1 Linear Programs Under Objective Uncertainty: A Completely Positive Representation

Abstract

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 are 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.

Copositive optimization – Recent developments and applications

“…In a similar domain [105] provide a completely positive formulation. They consider a mixed-binary linear optimization problem with stochastic objective function of which only the first two moments are known.…”

“…In general, it is difficult to test this condition, but it is satisfied if 1 µ ⊤ µ Σ is in the interior of C. Further assumptions are as in the previous section (Ax = b and x ∈ R n + imply 0 ≤ x j ≤ 1 for all j ∈ B), along with boundedness of the inner feasible set x ∈ R n + ∩ {0, 1} B n : Ax = b , which ensures that the expected value is bounded and thus z * as defined in (11) is finite. Then it is shown in [105] that…”

“…This recent development possibly opens new bridges from the SDP approaches of Cross Moment Models (CMM) and chance constraining via generalized Chebyshev boundsà la [124] to copositive optimization and its higherorder relaxations. For this reason, the authors of [105] suggest to call problem (12) Completely Positive Cross Moment Model (CPCMM). They also show an approximate equivalence result similar to those in the previous sections: for any optimal solution (x * , X * , Z * ) of (12), they construct a sequencec k of random vectors (with discrete distributions on R n + ) such that Ec k → µ and E c kc…”

Copositive Programming – a Survey

“…For a comment on Burer's result see [13]. Natarajan et al [45] consider (5) in the setting where Q = 0 and c is a random vector, and derive a completely positive formulation for the expected optimal value.…”

Extended trust-region problems with one or two balls: exact copositive and Lagrangian relaxations

“…Copositive optimization is a special case of convex conic optimization (namely, to minimize a linear function over a cone subject to linear constraints). By now, equivalent copositive reformulations for many important problems are known, among them (non-convex, mixed-binary, fractional) quadratic optimization problems under a mild assumption [2,3,13], and some special optimization problems under uncertainty [4,18,32,37]. In particular, it has been shown in [7] that, for quadratic optimization problems with additional nonnegative constraints, copositive relaxations (and its tractable approximations) provides a tighter bound than the usual Lagrangian relaxation.…”