Mixed integer programming approaches for nonlinear and stochastic programming

In this thesis we study how to solve some nonconvex optimization problems by using methods that capitalize on the success of Linear Programming (LP) based solvers for Mixed Integer Linear Programming (MILP). A common aspect of our solution approaches is the use, development and analysis of small but strong extended LP/MILP formulations and approximations.;In the first part of this work we develop an LP based branch-and-bound algorithm for mixed integer conic quadratic programs. The algorithm is based on a higher dimensional or lifted polyhedral relaxation of conic quadratic constraints introduced by Ben-Tal and Nemirovski. The algorithm is different from other LP based branch-and-bound algorithms for mixed integer nonlinear programs in that, it is not based on cuts from gradient inequalities and it sometimes branches on integer feasible solutions. We test the algorithm on a series of portfolio optimization problems and show that it significantly outperforms commercial and open source solvers based on both linear and nonlinear relaxations.;In the second part we study the modeling of a class of disjunctive constraints with a logarithmic number of binary variables and constraints. Many combinatorial constraints over continuous variables such as SOS1 and SOS2 constraints can be interpreted as disjunctive constraints that restrict the variables to lie in the union of a finite number of specially structured polyhedra. Known mixed integer formulations for these constraints have a number of binary variables and extra constraints linear in the number of polyhedra. We give sufficient conditions for constructing formulations for these constraints with a number of binary variables and extra constraints logarithmic in the number of polyhedra. Using these conditions we introduce mixed integer binary formulations for SOS1 and SOS2 constraints that have a number of binary variables and extra constraints logarithmic in the number of continuous variables. We also introduce the first mixed integer binary formulations for piecewise linear functions of one and two variables that use a number of binary variables and extra constraints logarithmic in the number of linear pieces of the functions. We prove that the new formulations for piecewise linear functions have favorable tightness properties and present computational results showing that they can significantly outperform other mixed integer binary formulations.;In the third part we study the modeling of non-convex piecewise linear functions as MILPs. We review several new and existing MILP formulations for continuous piecewise linear functions with special attention paid to multivariate non-separable functions. We compare these formulations with respect to their theoretical properties and their relative computational performance. In addition, we study the extension of these formulations to lower semicontinuous piecewise linear functions.;Finally, in the fourth part we study the strength of MILP formulations for LPs with Probabilistic Constraints. We first study the strength of existing MILP formulations that only considers one row of the probabilistic constraint at a time. We then introduce an extended formulation that considers more than one row of the constraint at a time and use it to computationally compare the relative strength between formulations that consider one and two rows at a time.