Topics in linear and nonlinear discrete optimization

This work contributes to modeling, theoretical, and practical aspects of structured Mathematical Programming problems. Many real-world applications have nonlinear characteristics and can be modeled as Mixed Integer Nonlinear Programming problems (MINLP). Modern global solvers have significant difficulty handling large-scale instances of them. Several convexification and underestimation techniques were proposed in the last decade as a part of the solution process, and we join this trend. The thesis has three major parts. The first part considers MINLP problems containing convex (in the sense of continuous relaxations) and posynomial terms (also called monomials), i.e. products of variables with some powers. Recently, a linear Mixed Integer Programming (MIP) approach was introduced for minimization the number of variables and transformations for convexification and underestimation of these structured problems. We provide polyhedral analysis together with separation for solving our variant of this minimization subproblem, containing binary and bounded continuous variables. Our novel mixed hyperedge method allows to outperform modern commercial MIP software, providing new families of facet-defining inequalities. As a byproduct, we introduce a new research area called mixed conflict hypergraphs. It merges mixed conflict graphs and 0-1 conflict hypergraphs. The second part applies our mixed hyperedge method to a linear subproblem of the same purpose for another class of structured MINLP problems. They contain signomial terms, i.e. posynomial terms of both positive and negative signs. We obtain new facet-defining inequalities in addition to those families from the first part. The final part is dedicated to managing guest flow in Georgia Aquarium after the Dolphin Tales opening with applying a large-scale MINLP. We consider arrival and departure processes related to scheduled shows and develop three stochastic models for them. If demand for the shows is high, all processes become interconnected and require a generalized model. We provide and solve a Signomial Programming problem with mixed variables for minimization resources to prevent and control congestions.Ph.D

Semi-Continuous Cuts for Mixed-Integer Programming

We study the convex hull of the feasible set of the semi-continuous knapsack problem, in which the variables belong to the union of two intervals. Besides being important in its own right, the semi-continuous knapsack problem is a relaxation of general mixed-integer programming. We show how strong inequalities valid for the semi-continuous knapsack polyhedron canbederivedandusedinabranch-and-cut scheme for mixed-integer programming and problems with semi-continuous variables. We present computational results that demonstrate the e®ectiveness of these inequalities, which we call collectively semi-continuous cuts. Our computational experience also shows that dealing with semi-continuous constraints directly in the branch-andcut algorithm through a specialized branching scheme and semi-continuous cuts is considerably more practical than the \textbook " approach of modeling semi-continuous constraints through the introduction of auxiliary binary variables in the model

Semi-Continuous Cuts for Mixed-Integer Programming?

Abstract. We study the convex hull of the feasible set of the semicontinuous knapsack problem, in which the variables belong to the union of two intervals. Besides being important in its own right, the semicontinuous knapsack problem is a relaxation of general mixed-integer programming. We show how strong inequalities that are valid for the semi-continuous knapsack polyhedron canbederivedandusedascutsin a branch-and-cut scheme for mixed-integer programming and problems with semi-continuous variables. We present computational results that demonstrate the e®ectiveness of these inequalities,which we call collectively semi-continuous cuts. Our computational experience also shows that dealing with semi-continuous constraints directly in the branchand-cut algorithm through a specialized branching scheme and semicontinuous cuts is considerably more practical than the \textbook " approach of modeling semi-continuous constraints through the introduction of auxiliary binary variables in the model