Pareto Optimality

Abstract

The Pareto optimality is based on the concept of dominance which definitions and properties are proposed. We distinguish weakly and strongly Pareto-optimal sets. The dominance binary relation is a strict partial order relation. This approach allows a comparison between feasible solutions in the objective space and the decision space. The nondominated solution sets yield Pareto fronts. Different methods are proposed to find good approximations of the Pareto sets when a Pareto front cannot be determined analytically. Numerous examples from the literature show connected and disconnected Pareto-optimal fronts in both decision space and fitness space. In particular, we can observe that objectives are conflicting, that the shapes of the Pareto front may be convex or nonconvex, connected or not, that Pareto fronts change if we decide to maximize instead to minimize the objectives. Necessary and sufficient conditions for Pareto optimality for constrained multi-objective optimization problems are also outlined.

Keywords: Conflicting objectives, Complementary slackness conditions, Dominance relation, Engineering design, Globally Pareto optimality, Ideal objective, KKT optimality conditions, KKT sufficient conditions, Locally Pareto optimality, Nadir point, Nonconflicting objectives, Nondominated solutions, Order relation, Pareto-optimal front, Partial polar cone, Slater constraint qualification, Strongly Pareto optimality, Trade off, Weakly Pareto optimality.