Extensions of the Gauvin-Tolle Optimal Value Differential Stability Results to General Mathematical Programs.

Abstract

Gauvin and Tolle have obtained bounds on the direc t ional derivative limit quotient of the optima l value func t ion for mathematical programs containing a ri ght-hand side perturbation. In this paper , we extend the results of Gauvin and Tolle to the general mathematical program in w ' uich a parameter appears arbitraril y in the constraints and in the objective function. An imp licit function theorem is app lied to transform the general mathema t ical program to a locally equivalent inequality constrained program , and , under conditions used by Gauvin and Tolle , their upper and lower bounds on the optimal

Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming

“…For a direct approach to such parameters, cf. [12] and related work of Fiacco and Hutzler [ 10]. The infinite-dimensional case too has been studied to a certain extent [21][22][23]14].…”

Optimal Value Differential Stability Bounds under the Mangasarian-Fromovitz Constraint Qualification.

“…The remaining results, Lemmas 3.8, 3.10, 4.3, 4.4 and Theorems 3.9, 3.11, 4.6, 4.8, and 4.9, extend the reduction approach introduced Jn [10] and complete the development of the theory for the general inequality-equality constrained parametric problem.…”

“…The purpose of this paper is to refine and continue the preliminary but incomplete study conducted by Fiacco and Rutzler [10] that extends the result-3 of Gauvin and Tolle 113) to the general inequality constrained mathematical program in which a parameter appears arbitrarity in the constraints and the objective function. We compl.ete this extension and…”

“…In a paper essentially 'simultaneous to [10], Gauvin and Dubeau [12] have independently obtained the differential boundn under the same assumptions invoked here but using a different method of proof.…”

“…Sections 1 through 3, through Theorem 3.7, and Sections 5 and 6 are taken more or less intact from [10] and are reported here for completeness.…”