A Geometric Approach to Integer Optimization and Its Application for Reachability Analysis in Petri Nets - Gu, Shenshen (ProQuest Dissertations & Theses, 2009)
Integer programming plays an important role in operations research and has a wide range of applications in various fields. There are a lot of research directions in the area of integer programming. In this thesis, two main topics will be investigated in details. One is to find the optimal binary solution to a quadratic object function, and the other is to find integer solutions to linear equations. Finding the optimal binary solution to a quadratic object function is known as the Binary Quadratic Programming problem (BQP), which has been studied extensively in the last three decades. In this thesis, by investigating geometric features of the ellipse contour of a concave quadratic function, we derive new upper and lower bounding methods for BQP. Integrating these new bounding schemes into a proposed solution algorithm of a branch-and-bound type, we propose an exact solution method in solving general BQP with promising preliminary computational results. Meanwhile, by investigating some special structures of the second order matrix and linear term in BQP, several polynomial time algorithms are discussed to solve some special cases of BQP. In the realm of integer programming, finding integer solutions to linear equations is another important research direction. The problem is proved to be NP-Complete, and several algorithms have been proposed such as the algorithm based on linear Diophantine equations as well as the method based on Groebner bases. Unlike the traditional algorithms, the new efficient method we propose in this thesis is based on our results on zero duality gap and the cell enumeration of an arrangement of hyperplanes in discrete geometry. Finding integer solutions to linear equations has various real world applications. In the thesis, we investigate its application to the reachability analysis of Petri nets. Introduced by Petri in 1962, Petri net has been a powerful mathematical formalism for modeling, analyzing and designing discrete event systems. In the research community of Petri nets, finding a feasible path from the initial state to the target state in Petri net, known as reachability analysis, is probably one of the most important and challenging subjects. The reachability algebraic analysis is equivalent to finding the nonnegative integer solutions to a fundamental equation constructed from the Petri net. We apply our algorithm in this thesis to reachability analysis of Petri net by finding the nonnegative integer solutions to the fundamental equation.