Strong Partitioning and a Machine Learning Approximation for Accelerating the Global Optimization of Nonconvex QCQPs (2301.00306v3)

Abstract: We learn optimal instance-specific heuristics for the global minimization of nonconvex quadratically-constrained quadratic programs (QCQPs). Specifically, we consider partitioning-based convex mixed-integer programming relaxations for nonconvex QCQPs and propose the novel problem of strong partitioning to optimally partition variable domains without sacrificing global optimality. Since solving this max-min strong partitioning problem exactly can be very challenging, we design a local optimization method that leverages generalized gradients of the value function of its inner-minimization problem. However, even solving the strong partitioning problem to local optimality can be time-consuming. To address this, we propose a simple and practical ML approximation for homogeneous families of QCQPs. We conduct a detailed computational study on randomly generated QCQP families, including instances of the pooling problem, using the open-source global solver Alpine. Numerical experiments demonstrate that our ML approximation of strong partitioning reduces Alpine's solution time by a factor of 2 to 4.5 on average, with a maximum reduction factor of 10 to 200 across the different QCQP families.