Surrogate Programming and Multipliers in Quasi-convex Programming

A result due to D. Luenberger on the existence of multipliers in a quasiconvex programming problem is extended to the case of constraints given by an arbitrary convex cone under a constraint qualification condition more general than Slater’s condition. The existence of solutions is not assumed. We point out links with even convexity in the sense of Fenchel, quasisubdifferentiability in the sense of Greenberg-Pierskalla, and observe that the couples of primal-dual optimal solutions reduce to saddle points of a suitable Lagrangian function.