Multiple-objective optimization by a multiplier method of proper equality constraints--Part I: Theory

Lin [1] has shown that general multiple-objective (MO) optimization problems can be solved by transforming them into single-objective (SO) optimization problems with all but one of the multiple objectives converted to proper equality constraints. This paper continues the development of such a general-purpose MO optimization method. It is shown here that Lagrange multipliers, which are intermediate by-products in the process of solving SO constrained optimization problems (by analytical or numerical methods), can be effectively utilized for testing/verifying whether the objective-converted equality constraints are proper or not. Several useful necessary and sufficient conditions for properness are derived and expressed in terms of Lagrange multipliers. These necessary and sufficient conditions are derived using no convexity or concavity assumptions. They are suitable for general analytical characterization of Pareto-optimal solutions, as well as for numerical generation of Pareto-optimal solutions by penalty methods, primal-dual methods, and Hestenes-Powell-Rockafellar methods of multipliers.