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.
