Novel Relaxation Techniques for Global Optimization of NLPs and MINLPs

Abstract

Over the last three decades rapid advances in computer technology coupled with new theoretical developments have led to significant progress in deterministic global optimization. While this progress has been considerable, this remains a relatively underdeveloped area, as there exists many classes of problems that global optimization algorithms are unable to solve to global optimality.In this thesis, we present computational methodologies aimed at improving the performance of branch-and-bound-based global optimization algorithms. To this end, we propose novel relaxation and domain partitioning strategies for various classes of nonconvex nonlinear programs (NLPs) and mixed-integer nonlinear programs (MINLPs).In the first part of this thesis, we consider nonconvex optimization problems containing convex-transformable functions. We introduce a new class of cutting planes derived by exploiting convex-transformability of intermediate expressions of factorable programs. In the second part of this thesis, we turn our attention to nonconvex mixed-integerquadratic programs (MIQPs). We present a family of convex quadratic relaxations derived by convexifying nonconvex quadratic functions through uniform diagonal perturbationsof the quadratic matrix. We investigate the theoretical properties of these quadratic relaxations and show that they are equivalent to some particular semidefinite programs. Wealso introduce novel branching variable selection strategies which can be used in conjunction with the proposed quadratic relaxations. In the third part of this thesis, we consider a related class of convex quadratic relaxations.In particular, we propose a new class of quadratically constrained programming (QCP) relaxations derived via convex quadratic cuts obtained from non-uniform diagonal perturbations of the quadratic matrix. We show that these relaxations are an outer-approximation of a semi-infinite convex program which under certain conditions is equivalent to a wellknown semidefinite program relaxation.To demonstrate the computational benefits of the ideas investigated in this thesis, we implement the proposed relaxation and domain partitioning strategies into the state-of-the art global optimization solver BARON. We test our implementation by conducting extensive computational studies on a variety of nonconvex problems. Results demonstrate that, for many test problems, the proposed techniques lead to order-of-magnitude speedups,resulting in a new version of BARON which outperforms other widely used global optimization solvers.