Treffer: Duality Principles for Optimization Problems Dealing with the Difference of Vector-Valued Convex Mappings: Duality principles for optimization problems dealing with the difference of vector-valued convex mappings

Consider the following equivalent problems: \[ \min f(x)+g\bigl( G(x)-H(x) \bigr), \text{ on a vectorial space},\tag{P} \] and \[ \min f(x), \text{ for } G(x)-H(x)\leq 0,\tag{R} \] where \(f,g\) are convex functions and \(G,H\) are vector-valued mappings that are convex with respect to a partial vectorial order for which \(g\) is nondecreasing. In this paper the author obtains duality formulas for problems (P) and (R) in terms of the Legendre-Fenchel conjugates of the data functions. The author also provides relations between the optimal solutions of primal and dual problems and a general necessary optimality condition. In particular the author applies the results to the problem of minimization of the composite of a convex mapping with a nonincreasing convex function and the minimization of the upper envelope of a family of concave functions.