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:
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C.1.
The Mangasarian-Fromovitz constraint qualification is satisfied at every point of M[H, G].
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C.2.
Every Kuhn-Tucker point of P(f, H, GG) is strongly stable in the sense of Kojima.
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C.3.
Different Kuhn-Tucker points have different (f-)values.
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References
Guddat, J. and Jongen, H. Th. (1987), Structural Stability in Nonlinear Optimization, Optimization 18, 617–631.
Hirsch, M. W. (1976), Differential Topology, Springer-Verlag, Berlin, Heidelberg, New York.
Jongen, H. Th., Jonker, P., and Twilt, F. (1986), Nonlinear Optimization in 64–1, II. Transversality, Flows, Parametric Aspects, Peter Lang Verlag, Frankfurt a.M., Bern, New York.
Gauvin, J. (1977), A Necessary and Sufficient Regularity Condition to Have Bounded Multipliers in Nonconvex Programming, Mathematical Programming 12, 136–138.
Kojima, M. (1980), Strongly Stable Stationary Solutions in Nonlinear Programs, in S. M. Robinson (ed.) Analysis and Computation of Fixed Points, Academic Press, New York.
Guddat, J., Jongen, H. Th., and Rückmann, J. (1986), On Stability and Stationary Points in Nonlinear Optimization, J. Australian Math. Soc., Series B 28, 36–56.
Jongen, H. Th., Jonker, P., and Twilt, F. (1983), Nonlinear Optimization in 64–2, I. Morse Theory, Chebychev Approximation, Peter Lang Verlag, Frankfurt a.M., Bern, New York.
Jongen, H. Th., Twilt, F., and Weber, G.-W. (1989), Semi-Infinite Optimization: Structure and Stability of the Feasible Set, Memorandum No. 838, Universiteit Twente. To appear in Journal of Optimization Theory and Applications.
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Jongen, H.T., Weber, G.W. Nonlinear optimization: Characterization of structural stability. J Glob Optim 1, 47–64 (1991). https://doi.org/10.1007/BF00120665
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DOI: https://doi.org/10.1007/BF00120665