Algorithmic approximation of optimal value differential stability bounds in nonlinear programming
In this paper, we first consider equality constrained mathematical programs and determine conditions under which the optimal value directional derivative can be calculated using only first-order information about the problem functions. We also give conditions under which this directional derivative can be estimated using the iterates of any sequential solution technique. Next we consider the more general program containing inequality as well as equality constraints. Using a mixed interior-exterior penalty function, we show that, when the parameter directional derivative of the optimal value function exists, it can be approximated or bounded above, depending on the nature of the solution generated by the penalty function algorithm. Moreover, we establish the existence of the parameter directional derivative of the mixed interior-exterior penalty function and obtain a representation of it.