The Equivalence And Generalization Of Optimization Criteria

Abstract

In this dissertation we first show that existing optimization criteria are equivalent to the maximization of a real-valued function in a one-dimensional Euclidean space. The criteria are said to be scalar equivalent. All solutions and only solutions to an optimization problem involving the original criterion can be obtained by scalarization without the typical convexity or concavity assumptions on the original objective functions and feasible region. Examples include Pareto (including the scalar case), satisficing, maximin, and cone-ordered optimization, as well as the more general notion of set-valued optimization in abstract spaces. Moreover, equivalences between various different optimization criteria are also established directly. As a consequence, any problem stated as one criterion can be solved as another. Second, we axiomatize and generalize the definition of an optimization criterion definition to include the existing standard criteria as special cases. We discuss our choices of axioms and explain why other possible axioms are excluded from our formalization. We then propose an equivalent scalarization of a general optimization criterion problem. In other words, we can obtain solutions of a problem involving any criterion satisfied our definition by simply solving scalar maximization problems. We present examples of new optimization criteria and apply them in practical decision-making situations. In addition, to provide insight into the scope of our work, we give a decision rule that is not a criterion within our framework.