Least Square Methods for Solving Systems of Inequalities With Application to an Assignment Problem

This research addresses algorithmic approaches for solving two different, but related, types of optimization problems. Firstly, the research considers the solution of a specific type of assignment problem using continuous methods. Secondly, the research addresses solving systems of inequalities (and equalities) in a least square sense. The specific assignment problem has piece-wise linear additive separable server cost functions, which are continuous everywhere except at zero, the point of discontinuity for the $\{0,1\}$ assignment condition. Continuous relaxation of the $\{0,1\}$ constraints yields a linear programming problem. Solving the dual of the linear programming problem yields the complementarity conditions for a primal solution, a system of linear inequalities and equalities. Adding equations to this system to enforce a $\{0,1\}$ solution in the relaxed solution set yields an augmented system, not necessarily linear. Methods to solve this system, a system of linear inequalities and non-linear equations, in a least square sense are developed, extending Han's method for solving linear systems of inequalities. Generalizations of these methods to solve general systems of inequalities in a least square sense are developed. The specific assignment problem is a variation of problems which are amenable to strong continuous relaxation, in that the solution set of the relaxed problem has been shown, experimentally, to often contain a $\{0,1\}$ solution. However, if there are a large number of variables, efficient continuous (non-combinatoric) methods are needed to locate $\{0,1\}$ solutions, if such exist. This work addresses methods to find $\{0,1\}$ solutions using a least square formulation for solving systems of inequalities.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2014

Author dc:creator

• Spoonamore, Janet Hurst

Contributors dc:contributor

• Bramley, R.,

• Mathematics
• Computer Science

Identifier

(UMI)AAI9305702

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/72536