## Date of Award

12-2012

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Legacy Department

Mathematical Science

## Committee Chair/Advisor

Adams, Warren P

## Committee Member

Wiecek , Margaret M

## Committee Member

Saltzman , Matthew J

## Committee Member

Belotti , Peitro L

## Abstract

This research is concerned with developing improved representations for special families of mixed-discrete programming problems. Such problems can typically be modeled using different mathematical forms, and the representation employed can greatly influence the problem's ability to be solved. Generally speaking, it is desired to obtain mixed 0-1 linear forms whose continuous relaxations provide tight polyhedral outer-approximations to the convex hulls of feasible solutions. This dissertation makes contributions to three distinct problems, providing new forms that improve upon published works.

The first emphasis is on devising solution procedures for the classical quadratic semi-assignment problem(QSAP), which is an NP-hard 0-1 quadratic program. The effort begins by using a reformulation-linearization technique to recast the problem as a mixed 0-1 linear program. The resulting form provides insight into identifying special instances that are readily solvable. For the general case, the form is shown to have a tight continuous relaxation, as well as to possess a decomposable structure. Specifically, a Hamiltonian decomposition of a graph interpretation is devised to motivate a Lagrangian dual whose subproblems consist of families of separable acyclic minimum-cost network flows. The result is an efficient approach for computing tight lower bounds on the optimal objective value to the original discrete program. Extensive computational experience is reported to evaluate the tightness of the representation and the expedience of the algorithm.

The second contribution uses disjunctive programming arguments to model the convex hull of the union of a finite collection of polytopes. It is well known that the convex hull of the union of n polytopes can be obtained by lifting the problem into a higher-dimensional space using n auxiliary continuous (scaling) variables. When placed within a larger optimization problem, these variables must be restricted to be binary. This work examines an approach that uses fewer binary variables. The same scaling technique is employed, but the variables are treated as continuous by introducing a logarithmic number of new binary variables and constraints. The scaling variables can now be substituted from the problem. Moreover, an emphasis of this work, is that specially structured polytopes lead to well-defined projection operations that yield more concise forms. These special polytopes consist of knapsack problems having SOS-1 and SOS-2 type restrictions. Different projections are defined for the SOS-2 case, leading to forms that serve to both explain and unify alternative representations for piecewise-linear functions, as well as to promote favorable computational experience.

The third contribution uses minimal cover and set covering inequalities to define the previously unknown convex hulls of special sets of binary vectors that are lexicographically lower and upper bounded by given vectors. These convex hulls are used to obtain ideal representations for base-2 expansions of bounded integer variables, and also afford a new perspective on, and extend convex hull results for, binary knapsack polytopes having weakly super-decreasing coefficients. Computational experience for base-2 expansions of integer variables exhibits a reduction in effort.

## Recommended Citation

Muldoon, Frank, "Polyhedral Approximations of Quadratic Semi-Assignment Problems, Disjunctive Programs, and Base-2 Expansions of Integer Variables" (2012). *All Dissertations*. 1054.

https://open.clemson.edu/all_dissertations/1054