Bounding the gap between the McCormick relaxation and the convex hull for bilinear functions

Abstract

We investigate how well the graph of a bilinear function b : [0, 1] n → R can be approximated by its McCormick relaxation. In particular, we are interested in the smallest number c such that the difference between the concave upper bounding and convex lower bounding functions obtained from the McCormick relaxation approach is at most c times the difference between the concave and convex envelopes. Answering a question of Luedtke, Namazifar and Linderoth, we show that this factor c cannot be bounded by a constant independent of n. More precisely, we show that for a random bilinear function b we have asymptotically almost surely c √ n/4. On the other hand, we prove that c 600 √ n, which improves the linear upper bound proved by Luedtke, Namazifar and Linderoth. In addition, we present an alternative proof for a result of Misener, Smadbeck and Floudas characterizing functions b for which the McCormick relaxation is equal to the convex hull.

The Multilinear Polytope for Acyclic Hypergraphs

“…, n} appears in at least one multilinear term. Building convex relaxations for Multilinear sets has been a subject of extensive research by the mathematical programming community [1,36,19,38,41,39,34,3,33,23,22,9,20,11]. Throughout this paper, we refer to the convex hull of the Multilinear set as the Multilinear polytope (MP).…”

“…Let us consider the inequalities in the description of MP LP G given by (3). Inequalities z v ≤ 1, for every v ∈ V , are given by (9). Inequalities z e ≥ 0, for every e such that e is not contained in any other edge are given by (10).…”

Strong formulations for quadratic optimization with M-matrices and indicator variables

“…Convex relaxations of S can also be constructed from the mixed-integer epigraph of the bilinear function ∑ i = j Q i j y i y j . There is an increasing amount of recent work focusing on bilinear functions [e.g., 13,14,35]. However, the convex hull of such functions is not fully understood even in the continuous case.…”

Error bounds for monomial convexification in polynomial optimization

“…They showed that for every x ∈ [0, 1] n , the ratio of the difference between the McCormick overestimator and underestimator values at x and the difference between the concave and convex envelope values at x can be bounded by a constant that is solely in terms of the chromatic number of the co-occurrence graph of the bilinear polynomial. Recently, Boland et al [Bol+17] showed that this same ratio cannot be bounded by a constant independent of n. Another, and somewhat natural, way of measuring the error from a relaxation is to bound the absolute gap z * S −z S , wherez S is a lower bound on z * S due to some convex relaxation of {(x, w) ∈ S × R | w = p(x)}. Such a bound helps determine how close one is to optimality in a global optimization algorithm.…”