Abstract:

During the last few decades, multiobjective programming has received much attention for both its numerous theoretical advances as well as its continued success in modeling and solving real-life decision problems in business and engineering. In extension of the traditional concept of Pareto optimality, this research investigates the more general notion of domination and establishes various theoretical results that lead to new optimization methods and support decision-making.

I first present some new results that characterize the nondominated set of a general vector optimization problem when the underlying domination structure is defined in terms of cones. Using concepts from linear algebra and convex analysis, a well known result relating nondominated points for polyhedral cones with Pareto solutions is generalized to nonpolyhedral cones, that are induced by positively homogeneous functions, and to translated polyhedral cones that are used to describe a notion of approximate nondominance. Pareto-oriented scalarization methods are modified and new solution methods are proposed for these two classes of cones. In addition, necessary and sufficient conditions for nondominance with respect to a variable domination cone are developed, and some specific results for the case of Bishop-Phelps cones are derived.

Based on my theoretical results, I then propose objective decompositions for large-scale multiobjective programs and establish relationships between solutions for the original and for the decomposed subproblems. Using the concept of approximate nondominance, a novel decision-making procedure is formulated and coordinates tradeoffs between these subproblems to find a preferred solution for the overall problem. The procedure is applied to a portfolio optimization problem.