Existence and sum decomposition of vertex polyhedral convex envelopes

Abstract

Convex envelopes are a very useful tool in global optimization. However finding the exact convex envelope of a function is a difficult task in general. This task becomes considerably simpler in the case where the domain is a polyhedron and the convex envelope is vertex polyhedral, i.e., has a polyhedral epigraph whose vertices correspond to the vertices of the domain. A further simplification is possible when the convex envelope is sum decomposable, i.e., the convex envelope of a sum of functions coincides with the sum of the convex envelopes of the summands. In this paper we provide characterizations and sufficient conditions for the existence of a vertex polyhedral convex envelope. Our results extend and unify several results previously obtained for special cases of this problem. We then characterize sum decomposability of vertex polyhedral convex envelopes, and we show, among else, that the vertex polyhedral convex envelope of a sum of functions coincides with the sum of the vertex polyhedral convex envelopes of the summands if and only if the latter sum is vertex polyhedral

GloMIQO: Global mixed-integer quadratic optimizer

“…The GloMIQO reformulation uses the observation that disaggregating bilinear terms tightens the relaxation of MIQCQP and actively takes advantage of any redundant linear constraints added to the model. It is standard to use termwise convex/concave envelopes [11,91] to relax MIQCQP, but many tighter relaxations have been developed based on: polyhedral facets of edge-concave multivariable term aggregations [17,26,34,94,95,96,99,111,130,131,132], eigenvector projections [38,106,113,122], piecewise-linear underestimators [29,65,66,73,93,98,99,100,101,107,119,139], outer approximation of convex terms [32,48,47], and semidefinite programming (SDP) relaxations [16,25,35,122,121]. GloMIQO incorporates several of these advanced relaxations.…”

“…Because separable functions are sum decomposable, the convex envelope of a sum of separable functions coincides with the sum of the convex envelopes of the separable functions [96,99,130,131,132]. In other words, finding the convex envelope of the equation represented by Figure 3(a) is equivalent to finding the convex envelopes of the four separable multivariable terms in Figure 3(b).…”

“…where q i, i , q j, j , q k, k are non-positive scalars, we used results of Tardella [130,131,132] and Meyer and Floudas [96] to generate the convex hull of (EC-AGG) and elucidate the facets of (EC-AGG) that strictly dominate the termwise relaxation of MIQCQP.…”

“…-We only augment the MILP relaxation of MIQCQP with eigenvector projections of multivariable terms that are not sum decomposable as defined by Tardella [130,131,132] and Meyer and Floudas [96]. Differently stated, we only integrate the eigenvector projection relaxation for multivariable terms where the convex envelope of the aggregate term may be different than the sum of the individual McCormick relaxations.…”

APOGEE: Global optimization of standard, generalized, and extended pooling problems via linear and logarithmic partitioning schemes

“…The MINLP formulation contains nonconvex constraints that include bilinear, multilinear, exponential, and power law terms which bear many similarities to the model of Furman and Androulakis [2008]. We developed a mixed-integer linear relaxation of the MINLP using piecewise-linear , Karuppiah and Grossmann, 2006, Meyer and Floudas, 2006, Wicaksono and Karimi, 2008, edge-concave , Tardella, 1988/89, 2003, 2008, and outer approximation relaxations. We integrated these relaxations into a branch-and-bound algorithm and solved several large-scale instances to global optimality .…”