ISSN:
1573-2916
Keywords:
90C31
;
Nonlinear optimization
;
structural stability
;
constraint qualification (Mangasarian-Fromovitz)
;
strong stability (Kojima)
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract We study global stability properties for differentiable optimization problems of the type: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaai% ikaGqaaiaa-jzacaGGSaGaamisaiaacYcacaqGGaGaam4raiaacMca% caGG6aGaaeiiaiaab2eacaqGPbGaaeOBaiaabccacaWFsgGaaeikai% aadIhacaqGPaGaaeiiaiaab+gacaqGUbGaaeiiaiaad2eacaGGBbGa% amisaiaacYcacaWGhbGaaiyxaiabg2da9iaacUhacaWG4bGaeyicI4% CeeuuDJXwAKbsr4rNCHbacfaGae4xhHe6aaWbaaSqabeaacaWGUbaa% aOGaaiiFaiaabccacaWGibGaaiikaiaadIhacaGGPaGaeyypa0JaaG% imaiaacYcacaqGGaGaam4raiaacIcacaWG4bGaaiykamaamaaabaGa% eyyzImlaaiaaicdacaGG9bGaaiOlaaaa!6B2E!\[P(f,H,{\text{ }}G):{\text{ Min }}f{\text{(}}x{\text{) on }}M[H,G] = \{ x \in \mathbb{R}^n |{\text{ }}H(x) = 0,{\text{ }}G(x)\underline \geqslant 0\} .\] Two problems are called equivalent if each lower level set of one problem is mapped homeomorphically onto a corresponding lower level set of the other one. In case that P(% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaaceWFsg% GbaGaacaWFSaGaa8hiaiqadIeagaacaiaacYcacaWFGaGabm4rayaa% iaaaaa!3EBF!\[\tilde f, \tilde H, \tilde G\]) is equivalent with P(f, H, GG) for all (% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaaceWFsg% GbaGaacaWFSaGaa8hiaiqadIeagaacaiaacYcacaWFGaGabm4rayaa% iaaaaa!3EBF!\[\tilde f, \tilde H, \tilde G\]) in some neighbourhood 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, GG) is equivalent with the validity of the following three conditions: C.1. The Mangasarian-Fromovitz constraint qualification is satisfied at every point of M[H, G]. C.2. Every Kuhn-Tucker point of P(f, H, GG) is strongly stable in the sense of Kojima. C.3. Different Kuhn-Tucker points have different (f-)values.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00120665
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