Strongly Stable Stationary Solutions in Nonlinear Programs

Abstract

This chapter focuses on strongly stable stationary solutions in nonlinear programs. Many studies have been made on the stability or the sensitivity of local minimum solutions to parametric programs. Those studies mainly discussed the continuity of the minimum value of the objective function, the continuity of the set of minimum solutions and/or the continuity of an isolated local minimum or stationary solution with respect to a small change of the parameter vector. In the case where the strict complementarity does not hold, the approach based on the standard implicit function theorem for continuously differentiable maps cannot be used. Various lemmas and theorems are discussed in the chapter. Stationary index, s-stable local minimum solutions, and degenerate s-stable stationary solutions are discussed. An application to a class of continuous deformation methods is reviewed.