ABSTRACT
We consider an ordinary convex program that does not necessarily attain its infimum. It is The continuity of the optimal value function and bounds on its upper and lower Dini derivatives are obtained for a general class of nonlinear parametric programs, using elementary and constructive arguments. An implicit function theorem is applied to transform a general parametric mathematical program into a locally equivalent inequality constrained program, and upper and lower bounds on the optimal value function directional derivative limit quotient are shown to hold for this reduced program. These bounds are then shown to apply in programs having both inequality and equality constraints where a parameter may appear anywhere in the program. This paper draws on several preliminary results reported by Fiacco and Hutzler for the inequality constrained problem and provides a number of extensions and missing proofs. These results generalize those provided by Gauvin and Tolle for the right-hand-side perturbation problem. Gauvin and Dubeau have essentially 66simultaneously obtained the same bounds using a completely different method of proof. Finally, it is noted that the continuity and bounds results for the general parametric program are immediate implications of the Gauvin-Tolle right-hand-side parameter results applied to an equivalent right-hand-side problem of higher dimension.
