Global Solution to Parametric Complementarity Constrained Programs and Applications in Optimal Parameter Selection By

This thesis contains five chapters. The notations, terminologies, definitions and numbering of equations, theorems and algorithms are independent in each chapter. Chapter 1 provides a fundamental introduction and contextual discussions to provide a unified theme for the subsequent chapters into a complete work. Chapters 2, 3 and 4 are arranged for ease of reading and understanding separately. Future research directions are proposed in Chapter 5 based on our findings. Chapter 1, Parametric Complementarity Constrained Programs– a Review of Methodologies, summarizes the basic techniques that are used in the algorithms for solving the mathematical program with complementarity constraints (MPCC), which is also referred to as the mathematical program with equilibrium constraints (MPEC) interchangeably in the chapter. We review the philosophy and main techniques behind the existing algorithms developed for solving MPEC. This background knowledge is followed by a section focusing on the methodologies for solving the specific class of problems that are uni-parametric, bi-parametric, and multiparametric complementarity constrained. One of the main sources of the parametric complementarity constrained program, inverse optimization, is defined in this chapter. A linear program with linear complementarity constraints (LPCC) is among the simplest mathematical programs with complementarity constraints. Yet the global solution of the LPCC remains difficult to find and/or verify. In Chapter 2, Global Solution of Bi-Parametric Linear Complementarity Constrained Linear Programs, we study a specific type of the LPCC which we term a bi-parametric LPCC. Reformulating the bi-parametric LPCC as a non-convex quadratically constrained program, we develop a domain-partitioning algorithm that solves a series of linear