Continuous deformation of nonlinear programs

Kojima, M.; Hirabayashi, Ryuichi

doi:10.1007/bfb0121217

Cited by 63 publications

(21 citation statements)

References 9 publications

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“…In 1980, M. Kojima introduced in [80] the (topological) concept of strong stability for stationary solutions (Karush-Kuhn-Tucker points) for nonlinear optimization problems (see also [100] by Robinson). This concept plays an important role in optimization theory, for example in sensitivity and parametric optimization [62,81], and structural stability [76]. It turns out that the concept of C-stationarity is the adequate stationarity concept regarding possible bifurcations.…”

Section: Parametric Aspectsmentioning

confidence: 99%

Topological Aspects of Nonsmooth Optimization

Shikhman

2012

Springer Optimization and Its Applications

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Section: Parametric Aspectsmentioning

confidence: 99%

Topological Aspects of Nonsmooth Optimization

Shikhman

2012

Springer Optimization and Its Applications

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“…This observation is in a certain sense connected with an interesting study in [16]. [12], where it is assumed that the Mangasarian-Fromowitz constraint qualification holds (cf. Section 4).…”

Section: M= {X Er"l H~(x) =0 Gj(x)>~o I E I J E J}mentioning

confidence: 87%

“…This set has been studied extensively in the important paper [12] under the additional assumption of the so-called Mangasarian-Fromowitz Constraint Qualification (shortly: MFCQ). However, the MFCQ is not a generic condition in optimization problems depending on parameters.…”

Section: The Kuhn-tucker Subsetmentioning

confidence: 99%

Critical sets in parametric optimization

Jongen

Jonker

Twilt

1986

Mathematical Programming

108

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We deal with one-parameter families of optimization problems in finite dimensions. The constraints are both of equality and inequality type. The concept of a 'generalized critical point' (g.c. point) is introduced. In particular, every local minimum, Kuhn-Tucker point, and point of Fritz John type is a g.c. point. Under fairly weak (even generic) conditions we study the set Z consisting of all g.c. points. Due to the parameter, the set X is pieced together from one-dimensional manifolds. The points of Z can be divided into five (characteristic) types. The subset of 'nondegenerate critical points' (first type) is open and dense in 2 (nondegenerate means: strict complementarity, nondegeneracy of the corresponding quadratic form and linear independence of the gradients of binding constraints). A nondegenerate critical point is completely characterized by means of four indices. The change of these indices along X is presented. Finally, the Kuhn-Tucker subset of 2; is studied in more detail, in particular in connection with the (failure of the) Mangasarian-Fromowitz constraint qualification.

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“…This implies (15). Consider the special height function O(x, t, A) = t. Then, (~, ?, 0) is a critical point for qbl~, and the corresponding Lagrange parameters a l , .…”

Section: ~=-O~ Dxfomentioning

confidence: 99%

One-parameter families of optimization problems: Equality constraints

Jongen

Jonker

1986

J Optim Theory Appl

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Abstract. In this paper, we introduce generalized critical points and discuss their relationship with other concepts of critical points [resp., stationary points]. Generalized critical points play an important role in parametric optimization. Under generic regularity conditions, we study the set of generalized critical points, in particular, the change of the Morse index. We focus our attention on problems with equality constraints only and provide an indication of how the present theory can be extended to problems with inequality constraints as well.

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Continuous deformation of nonlinear programs

Cited by 63 publications

References 9 publications

Topological Aspects of Nonsmooth Optimization

Topological Aspects of Nonsmooth Optimization

Critical sets in parametric optimization

One-parameter families of optimization problems: Equality constraints

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