Unbounded convex sets for non-convex mixed-integer quadratic programming

Abstract

This paper introduces a fundamental family of unbounded convex sets that arises in the context of non-convex mixed-integer quadratic programming. It is shown that any mixed-integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a set in the family. Some fundamental properties of the convex sets are derived, along with connections to some other well-studied convex sets. Several classes of valid and facet-inducing inequalities are also derived.

A note on representations of linear inequalities in non-convex mixed-integer quadratic programs

“…One can check that α Tx = β and α TX α = βL + U (β − L). Thus, (x,X) is feasible for the relaxation, but violates the inequality (5). All that remains is to show that the relaxation satisfies (6).…”

“…This relaxation satisfies (4). It does not in general satisfy (5), but it does satisfy the weaker inequality…”

“…This relaxation is however valid only if (i) b > 0 and (ii) all feasible MIQP solutions satisfy Ax ≥ −b. The analogue of SQK2 + is obtained by replacing each constraint α T x ≤ β with the stronger constraint (5). As for the analogue of SQK3, this is obtained by replacing each constraint α T x ≤ β with the 2n constraints presented in the following proposition.…”

“…The relaxation can be strengthened further by adding various valid inequalities, some of which exploit integrality; see, e.g., [5,6,13].…”

Non-convex mixed-integer nonlinear programming: A survey

“…Burer and Letchford [124] use a combination of polyhedral theory and convex analysis to analyze this convex set. In a followup paper, Burer and Letchford [126] apply the same approach to the case in which there are unbounded continuous and integer variables.…”