Certifiable Global Optimality for Large-Scale Nonconvex Mixed-Integer Programs

Mixed-Integer Nonlinear Programming (MINLP) problems are ubiquitous in science and engineering, modeling complex systems where decisions are both continuous and discrete, and relationships are nonlinear. However, the presence of nonconvexity and large-scale structures presents significant challenges to obtaining and, crucially, certifying global optimal solutions. Traditional methods often yield only local optima or struggle with computational tractability for larger instances. This paper addresses the critical need for robust and certifiable global optimality for large-scale nonconvex MINLPs. We present a novel framework that integrates advanced convex relaxation techniques, such as strong semidefinite programming relaxations and McCormick envelopes, with a hierarchical decomposition approach. The methodology combines spatial branch-and-bound strategies with cutting plane methods and primal-dual algorithms tailored for large-scale problems. A key aspect is the explicit construction of optimality certificates, providing a rigorous guarantee that the obtained solution is indeed globally optimal. Through extensive theoretical analysis and rigorous computational results on benchmark problems, our approach demonstrates significant tightening of lower bounds, substantial improvements in computational efficiency, and the successful attainment of verifiable global solutions for previously intractable large-scale nonconvex MINLPs. This work contributes to bridging the gap between theoretical global optimality guarantees and practical solvability for this challenging class of optimization problems.