Simultaneous Convexification of Bilinear Functions over Polytopes with Application to Network Interdiction

Abstract

We study the simultaneous convexification of graphs of bilinear functions $g^k({x};{y}) = {y}^{\intercal} A^k {x}$ over ${x} \in \ensuremath{\Xi} = \{ {x} \in [0,1]^n \, | \, E{x} \geq {f} \}$ and ${y} \in \Delta_m = \{ {y} \in {R}_+^{m} \, | \, {1^{\intercal}y} \leq 1 \}$. We propose a constructive procedure to obtain a linear description of the convex hull of the resulting set. This procedure can be applied to derive convex and concave envelopes of certain bilinear functions, to study unary expansions of integer variables in mixed integer bilinear sets, and to obtain convex hulls of sets with complementarity constraints. Exploiting the structure of $\Xi$, the procedure naturally yields stronger linearizations for bilinear terms in a variety of practical settings. In particular, we demonstrate the effectiveness of the approach by strengthening the traditional dual formulation of network interdiction problems and report encouraging preliminary numerical results.