Semidefinite Cuts and Partial Convexification Techniques with Applications to Continuous Nonconvex Optimization, Stochastic Integer Programming, and Facility Layout Problems

(ABSTRACT) Despite recent advances in convex optimization techniques, the areas of discrete and continuous nonconvex optimization remain formidable, particularly when globally optimal solutions are desired. Most solution techniques, such as branch-and-bound, are enumerative in nature, and the rate of their convergence is strongly dependent on the accuracy of the bounds provided, and therefore, on the tightness of the underlying formulation. This research develops both general and problem-specific procedures to be used in conjunction with the Reformulation-Linearization Technique (RLT) for generating tight model formulations for challenging nonconvex optimization problems. These problems include the general classes of nonlinear and integer programs, as well as specific applications within these areas. We begin by deriving a new class of cutting planes, called semidefinite cuts, for enhancing the solution of nonconvex optimization problems. While these cuts can be generally applied to either discrete or continuous nonconvex problems, we specifically demonstrate their effectiveness in solving quadratic optimization problems. We then focus on the important class of mixed-integer programming (MIP) problems, and develop a new decomposition technique. This methodology is particularly well-suited to solve stochastic integer programming problems, arguably, the most difficult class of discrete problems. Finally, we address a specific MIP application, known as the facility layout problem, that has defied exact solution methods, and which subsumes the notorious quadratic assignment problem. We significantly advance the state-of-the-art in solving these problems by developing substantially improved models and algorithms through outer-linearization techniques and concepts from disjunctive programming. Our first contribution proposes a mechanism to tighten RLT-based relaxations for general problems in nonconvex optimization by importing concepts from semidefinite programming (SDP), leading to a new class of semidefinite cutting planes. Given an RLT relaxation, the usual nonnegativity restrictions on the matrix of RLT product variables is replaced by a suitable positive semidefinite constraint. Instead of relying on specific SDP solvers, the positive semidefinite stipulation is rewritten to develop a semi-infinite linear programming representation of the problem, and an approach is developed that can be implemented using traditional optimization software. Specifically, the infinite set of constraints is relaxed, and members of this set are generated as needed via a separation routine in polynomial time. In essence, this process yields an RLT relaxation that is augmented with valid inequalities, which are themselves classes of RLT constraints that we call semidefinite cuts. We illustrate the use of this strategy by applying it to the case of optimizing a nonconvex quadratic objective …