STRONG VALID INEQUALITIES FOR MIXED-INTEGER NONLINEAR PROGRAMS VIA DISJUNCTIVE PROGRAMMING AND LIFTING By KWANGHUN CHUNG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy STRONG VALID INEQUALITIES FOR MIXED-INTEGER NONLINEAR PROGRAMS VIA DISJUNCTIVE PROGRAMMING AND LIFTING By Kwanghun Chung August 2010 Chair: Jean-Philippe. P. Richard Major: Industrial and Systems Engineering Mixed-Integer Nonlinear Programs (MINLP) are optimization problems that have found applications in virtually all sectors of the economy. Although these models can be used to design and improve a large array of practical systems, they are typically difficult to solve to global optimality. In this thesis, we introduce new tools for the solution of such problems. In particular, we develop new procedures to construct convex relaxations of certain MINLP problems. These relaxations are stronger than those currently known for these problems and therefore provide improvements in the solution of MINLPs through branch-and-bound techniques. There are three main components to our contributions. First, we derive a closed-form characterization of the convex hull of a generic nonlinear set, when the convex hull of this set is completely determined by orthogonal restrictions of the original set. Although the tools used in our derivation include disjunctive programming and convex extensions, our characterization does not introduce additional variables. We develop and apply a toolbox of results to check the technical assumptions under which this convexification tool can be employed. We demonstrate its applicability in integer programming by providing an alternate derivation of the split cut for mixed-integer polyhedral sets and by finding the convex hull of various mixed/pure-integer bilinear sets. We then develop a key result that extends the utility of the convexification tool to relaxing nonconvex inequalities, which are not naturally disjunctive, by providing sufficient conditions for establishing the convex extension