Multi-Objective Optimization: Lagrange Duality

As is well known, duality in mathematical programming is based on the property that any closed convex set can be also represented by the intersection of closed half spaces including it. Let the multi-objective optimization problem to be considered here be given by

where

Note here that vector inequalities are commonly used: for any n-vectors a and b, a > b means a _i > b _i (i = 1,..., n). Also, means (i = 1,..., n). On the other hand, a ≥ b means but a ≠ b. Hereafter, vector inequalities such as will be used instead of (i = 1,..., m).

Defining a dual problem (D) in some appropriate way associated with the problem (P), our aim is to show the property min(P) = max(D). Here min(P) denotes the set of efficient points of the problem (P) in the objective function space R ^p, and similarly max(D) the one of the dual problem (D).

Unlike the usual mathematical programming, the optimal value of the primal problem (and the dual problem) are not necessarily determined uniquely in multi-objective...