Summary

This thesis proposes exact techniques to solve a broad class of larger-scale nonconvex (robust) optimization problems. More specific, in this thesis we consider nonconvex (robust) optimization problems in which the nonconvex components are sums of convex times convex functions. A broad class of problems can be written in this way, such as mixed binary linear optimization problems, nonconvex quadratic optimization problems, difference of convex problems, and optimization problems containing robust convex constraints. Moreover, these nonconvex optimization problems appear in a large number of applications, e.g., in supply chain, network security, resource allocation in healthcare or energy markets, and traffic flow management.
The main approach in this thesis exploits two techniques: first, we introduce a new relaxation technique called the Reformulation-Perspectification Technique (RPT), to obtain a convex approximation of the considered nonconvex optimization problem. RPT generates additional redundant nonconvex constraints from pairwise multiplication of the existing convex inequalities in the nonconvex optimization problem. Subsequently, it convexifies all nonconvex components in the nonconvex optimization problem as well as in the additional generated constraints by reformulating them in their perspective form and subsequently linearizing all product terms. Next, we develop a spatial Branch and Bound scheme, leveraging RPT, to obtain a global optimal solution.