A piecewise relaxation for quadratically constrained problems based on a mixed-radix numeral system

Abstract

Non-convex quadratically constrained problems frequently appear in chemical engineering when optimizing process networks. Some of these problems can be solved to global optimality by deterministic solvers like BARON and ANTIGONE that mostly use linear programming relaxations coupled with spatial branch and bound. An alternative is to rely on piecewise relaxations, which work by simultaneously partitioning the domain of one variable in every bilinear term and can be significantly tighter, even when setting the number of intervals in the partitions, N, to a small value. In this short note, we generalize the mixed-integer linear programming relaxation formulation from the multiparametric disaggregation technique, to benefit from a logarithmic partitioning scheme in a wide variety of settings. The idea is to select the optimal interval in the partition of a variable by using a mixed-radix numeral system, following the prime factorization of N.