A “joint + marginal” heuristic for 0/1 programs

Abstract

We propose a heuristic for 0/1 programs based on the recent “joint + marginal” approach of the first author for parametric polynomial optimization. The idea is to first consider the n-variable (x ₁, . . . , x _n ) problem as a (n − 1)-variable problem (x ₂, . . . , x _n ) with the variable x ₁ being now a parameter taking value in {0, 1}. One then solves a hierarchy of what we call “joint + marginal” semidefinite relaxations whose duals provide a sequence of polynomial approximations ${x_1\mapsto J_k(x_1)}$ that converges to the optimal value function J (x ₁) (as a function of the parameter x ₁). One considers a fixed index k in the hierarchy and if J _k (1) > J _k (0) then one decides x ₁ = 1 and x ₁ = 0 otherwise. The quality of the approximation depends on how large k can be chosen (in general, for significant size problems, k = 1 is the only choice). One iterates the procedure with now a (n − 2)-variable problem with one parameter ${x_2 \in \{0, 1\}}$ , etc. Variants are also briefly described as well as some preliminary numerical experiments on the MAXCUT, k-cluster and 0/1 knapsack problems.