Multiple-objective optimization by method of proper equality constraints using lagrange multipliers

Lin [1] has shown that all Pareto-optimal solutions of any multiple-objective (MO) optimization problems can be determined by transforming the MO problems into single-objectives (SO) optimization problems with all but one of the multiple objectives converted to proper equality constraints. 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 determining 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. A very useful convention and a simple notion of regularity are introduced.