J. Guddat

Structural stability in nonlinear optimization

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 ue56f([fbar],ℏ,g) is equivalent with ue56f(f,H,G) for all ([fbar],ℏ,g) in some neighborhood of (f. H. G) we call ue56f(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 ue56f(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 ue56f(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 structu...

Pathfollowing with Jumps in the Set of Local Minimizers and in the Set of Generalized Critical Points

We consider the one-parametric optimization problem P(t), t∈[0, 1], where P(t) is taken as in Section 1.1, namely P(t): n n

On Dialogue Algorithms for Linear and Nonlinear Vector Optimization from the Point of View of Parametric Optimization

min left{ {fleft( {x,t} right)left| {x in M} right.left( t right)} right},t in left[ {0,1} right],

Parametric optimization: Pathfollowing with jumps

n n(5.1.1) n nwhere n n

Parametric Optimization and Related Topics IV, series "Approximation and Optimization"

Mleft( t right) = left{ {x in {mathbb{R}^n}left| {{h_i}left( {x,t} right) = 0,i in I,{g_j}left( {x,t} right) leqslant 0,j in } right.J} right}

Pathfollowing of Curves of Local Minimizers

n n(5.1.2) n nwith n n