Global optimization of mixed-integer quadratically-constrained quadratic programs (MIQCQP) through piecewise-linear and edge-concave relaxations

Abstract

We propose a deterministic global optimization approach, whose novel contributions are rooted in the edge-concave and piecewise-linear underestimators, to address nonconvex mixed-integer quadratically-constrained quadratic programs (MIQCQP) to ε-global optimality. The facets of low-dimensional (n ≤ 3) edge-concave aggregations dominating the termwise relaxation of MIQCQP are introduced at every node of a branch-and-bound tree. Concave multivariable terms and sparsely distributed bilinear terms that do not participate in connected edge-concave aggregations are addressed through piecewise-linear relaxations. Extensive computational studies are presented for point packing problems, standard and generalized pooling problems, and examples from GLOBALLib [55].

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).…”

“…-Low-Dimensional Edge-Concave Aggregations: As discussed in our previous work [99], GloMIQO detects low-dimensional edge-concave aggregations for the purpose of generating dominant cuts ( §3.3.2) and reducing variable bounds ( §3.5.1).…”

“…Our recent work analyzed the conditions under which low-dimensional edge-concave functions introduce dominant cuts into the MILP relaxation of MIQCQP [99]. Given the function:…”

“…As described in our earlier work [99], the GloMIQO preprocessing detects low-dimensional edge-concave aggregations that may lead to dominant cuts in the MILP relaxation. Then, at every node of the branch-and-bound tree, GloMIQO updates the facets of (EC-AGG) and compares them with the termwise relaxation of (EC-AGG).…”