Structural Stability in Nonlinear Optimization

Abstract

This paper is concerned with global stability properties for differentiate optimization problems of the type: We introduce a natural equivalence concept for optimization problems, In this equivalence each lower level set of one problem is mapped homeomorphically onto a corresponding lower level set of the other one. In case that 𝔓([fbar],ℏ,g) is equivalent with 𝔓(f,H,G) for all ([fbar],ℏ,g) in some neighborhood of (f. H. G) we call 𝔓(f, H, G) structurally stable; the topology used takes derivatives up to order two into account. Under the assumption that M[H, G] is compact we prove that structural stability of 𝔓(f,H,G) implies the following three conditions: C1. The Mangasarian-Fromovitz constraint qualification is satisfied at every point of M[H, G]. C2. Every Kuhn-Tucker point of 𝔓(f,H,G) is strongly stable in the sense of Kojima. C3. Different Kuhn-Tucker points have different (f-) val es. On the other hand, again in case M[H,G] is compact, we conjecture that the conditions CI, C2 and C3 imply structural stability.