Using Two-Dimensional Projections for Stronger Separation and Propagation of Bilinear Terms

Abstract

One of the most fundamental ingredients in mixed-integer nonlinear programming solvers is the well-known McCormick relaxation for a product of two variables x and y over a boxconstrained domain. The starting point of this paper is the fact that the convex hull of the graph of xy can be much tighter when computed over a strict, non-rectangular subset of the box. In order to exploit this in practice, we propose to compute valid linear inequalities for the projection of the feasible region onto the x-y-space by solving a sequence of linear programs akin to optimization-based bound tightening. These valid inequalities allow us to employ results from the literature to strengthen the classical McCormick relaxation. As a consequence, we obtain a stronger convexification procedure that exploits problem structure and can benefit from supplementary information obtained during the branchand bound algorithm such as an objective cutoff. We complement this by a new bound tightening procedure that efficiently computes the best possible bounds for x, y, and xy over the available projections. Our computational evaluation using the academic solver SCIP exhibit that the proposed methods are applicable to a large portion of the public test library MINLPLib and help to improve performance significantly.

Implications, conflicts, and reductions for Steiner trees

“…If v ∈ T , add the path P from v to S, marked by the predecessor array, to S, add V (P) to Q, and set d[u] := 0 for all u ∈ V (P). Furthermore, update (28). For all {v, w} ∈ δ(v) proceed as follows.…”

“…In this way, the distance associated with a path also reflects the cost needed to connect additional terminals later on. Note that the minimum spanning tree computed during postprocessing will always contain the edge associated with each vertex of positive implied profit contained in S. We use the value min e ∈δ(w)\{e} c(e ) instead of b(e) for e = {v, w}, w ∈ T in (28) for two reasons: First, the value better represents the weight that can be saved when connecting w via v (because the bottleneck edge corresponding to b(e) might already be part of the tree computed by the heuristic so far). Second, this value is much faster to compute (and the primal heuristic is executed often as a subroutine within our implementation).…”

“…Any group [10 k , 86,400] contains each instance from Table 3 such that [37] or SCIP-Jack solves this instance in not less than 10 k , and at most 86,400 s. If an instance can be solved by only one solver within the time-limit, we consider the run-time of the other solver on this instance as 86,400 s. Such groupings are commonly used in computational mathematical optimization (also with the time lower bounds being powers of 10), see e.g. [28,49]. In addition to the shifted geometric mean, Table 4 also provides the arithmetic mean of the run-time for each group.…”

“…Define a distance array d and a predecessor array pr ed by d[u] := ∞, pr ed[u] := null for all u ∈ V \{v 0 }, and d[v 0 ] := 0, pr ed[v 0 ] := v 0 . Define for all v ∈ V \T : p(v) := max 0, sup b(e) − c(e) | e = {v, w} ∈ δ(v), w ∈ T \V (S) (28).…”

Convex envelopes for ray-concave functions

“…This procedure was refined in [14], by shifting the calculations to the solution of a KKT system. These last techniques were extensively tested in [21] to improve general-purpose optimization routines. In [13], the author characterizes the convex envelope of various bivariate functions (including the bilinear and fractional functions) over arbitrary polytopes using a polyhedral sub-division of the polytopes.…”