Many decision problems arising from real-world applications can be modeled as disjunctive convex programs, i.e., optimization problems with a convex objective function and a feasible region defined as the union of individual convex sets having a known algebraic representation. This particular class of problems subsumes, among others, mixed-integer programs, linear complementarity problems and chance constrained problems (of finite support distribution). Examples of applications that can be modeled as disjunctive convex program with some nonlinear element in the formulation include the market product positioning problem (from Marketing), the process synthesis network design problem (from Chemical Engineering) and the limited-diversification portfolio selection problem (from Finance).
This dissertation is primarily concerned with the solution of the nonlinear models using cutting-plane based algorithms. Our work builds upon the theory and practical experience developed by Balas, Ceria and Cornuéjols with their “lift-and-project” method, which is designed to tackle mixed 0-1 linear programs-a subclass of disjunctive convex programs.
We first provide an unified treatment for the primal and dual representation of the convex hull of the union of individual polyhedra. Then, we focus on the lift-and-project cut generation problem to derive some new theoretical properties and introduce some new classes of cuts. Our primal-dual framework also allows for a differentiated geometric interpretation of lift-and-project cuts. We also show that a specialized version of the cutting-plane algorithm that uses cuts based on the euclidean norm has finite convergence.
In the nonlinear setting, we provide a primal representation of the convex hull of the union of individual convex sets, extending an earlier result of Jeroslow. Then, we propose an extension of the lift-and-project approach to tackle mixed 0-1convex programs. We explain that the cut generation problem can be solved in a smaller dimension space by a primal procedure akin to the interior-point approach or a primal-dual procedure which is a specialized form of the Bundle method, or Frank-Wolfe's algorithm.