Improving Solvability of Continuous and Mixed Integer Convex Optimization Problems Using Interior Approaches - Huang, Kuo-Ling (ProQuest Dissertations & Theses, 2012)
In this thesis we present a homogeneous interior point solver for linear and convex programming that is equipped with a continuously differentiable potential function. Our work is motivated by the apparent gap between the theoretical complexity results and long-step practical implementations in interior point algorithms. The potential function described in this thesis ensures a global linear polynomial-time convergence while providing the flexibility to integrate heuristics for generating the search directions and step length computations. A practical algorithm based on this potential function is implemented in a software package named iOptimize. When compared with a mature software package MOSEK (barrier solver version 6.0.0.106), iOptimize solves linear programming, convex quadratic programming problems, convex quadratically constrained quadratic programming problems, and general convex programming problems in fewer iterations. Moreover, several problems for which MOSEK fails are solved to optimality. We also present our computational experience in combining geometric random walks and the feasibility pump to heuristically generate feasible (possibly optimal) solutions for the mixed integer linear and convex optimization problems. Computational results on mixed integer linear optimization test libraries (MIPLIB2003 library and COR@L library) and a mixed integer convex optimization test library (CMU-IBM library) show that the walk-and-round approach improves the upper bounds of a large number of test problems when compared to running the feasibility pump either at the optimal solution or the analytic center of the continuous relaxation with similar computational efforts.