Lagrange Duality and Partitioning Techniques in Nonconvex Global Optimization

It is shown that, for very general classes of nonconvex global optimization problems, the duality gap obtained by solving a corresponding Lagrangian dual in reduced to zero in the limit when combined with suitably refined partitioning of the feasible set. A similar result holds for partly convex problems where exhaustive partitioning is applied only in the space of nonconvex variables. Applications include branch-and-bound approaches for linearly constrained problems where convex envelopes can be computed, certain generalized bilinear problems, linearly constrained optimization of the sum of ratios of affine functions, and concave minimization under reverse convex constraints.