Topics in linear, dynamic and multi-objective optimization

Zaili Zhang, Purdue University

Abstract

This thesis comprises three parts. The first part discusses the Gravitational method for Linear Programming. In the Gravitational method, a point particle is dropped from an interior feasible point of the polyhedron and at each instant it moves along the steepest feasible improving direction, called the feasible gradient. There are only finitely many feasible gradients for any linear program and it is shown that the magnitudes of the feasible gradients computed in the Gravitational method form a strictly monotonically decreasing sequence. Subsequently, the worst case time complexity of the Gravitational method is shown to be exponential in the size of the input Linear Program. The second part resolves two long-standing conjectures in multiple objective optimization, proposed by Soland. The conjectures concern relations between the sets of efficient and properly efficient solutions and the existence of value functions satisfying certain properties. We present some sufficient conditions for the validity of Soland's conjectures and show that the sufficient conditions cannot be relaxed. The third part studies relationships among the following important concepts in dynamic programming: strict monotonicity, monotonicity, Bellman's principle of optimality, Morin's principle of optimality, validity of functional equations and the ability to find an/all optimal solution(s) using functional equations.

Degree

Ph.D.

Advisors

Prabhu, Purdue University.

Subject Area

Industrial engineering

COinS