Thesis (M.S.)--University of Missouri--Rolla, 2007. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed December 7, 2007) Includes bibliographical references (p. 28).

In this thesis we consider four topics all related to using problem reformulations in order to solve integer programs, i.e. optimization problems in which the decision variables must be integer. We first consider the polyhedral approach. We start by addressing the question of lifting valid inequalities, i.e. finding a valid inequality for a set Y from the knowledge of a valid inequality for a lower-dimensional restriction X of Y. We simplify and clarify the presentation of the procedure. This allows us to derive conditions under which the computation of the lifting is tractable. The second topic is the study of valid inequalities for the single node flow set. The single node flow set is the problem obtained by considering one node of a fixed charge network flow problem. We derive valid inequalities for this set and various generalizations. Our approach is a systematic procedure using only basic tools of integer programming: fixing and complementing variables, mixed-integer rounding and lifting. The method allows us to explain and generate a large range of inequalities describing the convex hull of such sets. The last two topics are based on non-standard approaches for integer programming. We first show how the group relaxation approach can be used to provide reformulations for the integral basis method. This is based on a study of extended formulations for the group problem. We present four extended formulations and show that the projections of three of these formulations provide the convex hull of the original group problem. Initial computational tests of the approach are also reported. Finally we consider a problem that is difficult for the standard branch-and-bound approach even for small instances. A reformulation based on lattice basis reduction is known to be more effective. However the step to compute the reduced basis is O(n^4) and becomes a bottleneck for small to medium instances. By using the structure of the problem, we show that we can decompose the problem and obtain the basis by taking the kronecker product of two smaller bases easier to compute. Furthermore, if the two small bases are reduced, the kronecker product is also reduced up to a reordering of the vectors. Computational results show the gain from such an approach.

We study a generic minimization problem with separable non-convex piecewise linear costs, showing that the linear programming (LP) relaxation of three textbook mixed integer programming formulations each approximates the cost function by its lower convex envelope. We also show a relationship between this result and classical Lagrangian duality theory.

The United State’s renewable portfolio standards call for a large increase of renewable energy and improved conservation efforts over today’s current system. Wind will play a ma jor role in meeting the renewable portfolio standards. As a result, the amount of wind capacity and generation has been growing exponentially over the past 10 to 15 years. The proposed unit commitment method integrates wind energy into a scheduable resource while keeping the formulation simple using mixed integer programming. A reserve constraint is developed and added to unit commitment giving the forecasted wind energy an effective cost. The reserve constraint can be scaled based on the needs of the system: cost, reliability, or the penetration of wind energy. The results show that approximately 24% of the load can be met in the given test system, while keeping a constant reliability before and after wind is introduced. This amount of wind will alone meet many of the renewable portfolio standards in the United States.

Thesis (Ph. D.)--University of Chicago Graduate School of Business, December 1997. / Includes bibliographical references.

The mobility allowance shuttle transit (MAST) system is a hybrid transit system in which vehicles are allowed to deviate from a fixed route to serve flexible demand. A mixed integer programming (MIP) formulation for the static scheduling problem of a multi-vehicle Mobility Allowance Shuttle Transit (MAST) system is proposed in this thesis. Based on the MIP formulation, we analyze the impacts of time headways between consecutive transit vehicles on the performance of a two-vehicle MAST system. An analytical framework is then developed to model the performance of both one-vehicle and two-vehicle MAST systems, which is used to identify the critical demand level at which an increase of the fleet size from one to two vehicles would be appropriate. Finally, a sensitivity analysis is conducted to find out the impact of a key modeling parameter, w1, the weight of operations cost on the critical demand. In this paper, we develop an insertion heuristic for a multi-vehicle MAST system, which has never been addressed in the literature. The proposed heuristic is validated and evaluated by a set of simulations performed at different demand levels and with different control parameters. By comparing its performance versus the optimal solutions, the effectiveness of the heuristic is confirmed. Compared to its single-vehicle counterpart, the multiple-vehicle MAST prevails in terms of rejection rate, passenger waiting time and overall objective function, among other performance indices.

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This thesis presents novel approaches that use interior point concepts in solving mixed integer programs. Particularly, we use the analytic center cutting plane method to improve three of the main components of the branch-and-bound algorithm: cutting planes, heuristics, and branching. First, we present an interior point branch-and-cut algorithm for structured integer programs based on Benders decomposition. We explore using Benders decomposition in a branch-and-cut framework where the Benders cuts are generated using the analytic center cutting plane method. The algorithm is tested on two classes of problems: the capacitated facility location problem and the multicommodity capacitated fixed charge network design problem. For the capacitated facility location problem, the proposed approach was on average 2.5 times faster than Benders-branch-and-cut and 11 times faster than classical Benders decomposition. For the multicommodity capacitated fixed charge network design problem, the proposed approach was 4 times faster than Benders-branch-and-cut while classical Benders decomposition failed to solve the majority of the tested instances. Second, we present a heuristic algorithm for mixed integer programs based on interior points. As integer solutions are typically in the interior, we use the analytic center cutting plane method to search for integer feasible points within the interior of the feasible set. The algorithm searches along two line segments that connect the weighted analytic center and two extreme points of the linear programming relaxation. Candidate points are rounded and tested for feasibility. Cuts aimed to improve the objective function and restore feasibility are then added to displace the weighted analytic center until a feasible integer solution is found. The algorithm is composed of three phases. In the first, points along the two line segments are rounded gradually to find integer feasible solutions. Then in an attempt to improve the quality of the solutions, the cut related to the bound constraint is updated and a new weighted analytic center is found. Upon failing to find a feasible integer solution, a second phase is started where cuts related to the violated feasibility constraints are added. As a last resort, the algorithm solves a minimum distance problem in a third phase. For all the tested instances, the algorithm finds good quality feasible solutions in the first two phases and the third phase is never called. Finally, we present a new approach to generate good general branching constraints based on the shape of the polyhedron. Our approach is based on approximating the polyhedron using an inscribed ellipsoid. We use Dikin's ellipsoid which we calculate using the analytic center. We propose to use the disjunction that has a minimum width on the ellipsoid. We use the fact that the width of the ellipsoid in a given direction has a closed form solution in order to formulate a quadratic problem whose optimal solution is a thin direction of the ellipsoid. While solving a quadratic problem at each node of the branch-and-bound tree is impractical, we use a local search heuristic for its solution. Computational testing conducted on hard integer problems from MIPLIB and CORAL showed that the proposed approach outperforms classical branching.