Conditions of High Order for a Local Minimum in Problems With Constraints

Levitin, E. S.; Milyutin, A. A.; Osmolovskii, Nikolai P.

doi:10.1070/rm1978v033n06abeh003885

Cited by 95 publications

(58 citation statements)

References 26 publications

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“…(This philosophy is not new. It was clearly expressed by Warga [56] and, for a somewhat different purpose, by Levitin-Miljutin-Osmolovskii [62].) Most approaches to nonsmooth analysis can be considered special methods to construct prederivatives or define them axiomatically (as in [19]).…”

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confidence: 99%

Nonsmooth analysis: differential calculus of nondifferentiable mappings

Ioffe¹

1981

Trans. Amer. Math. Soc.

205

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Abstract.A new approach to local analysis of nonsmooth mappings from one Banach space into another is suggested. The approach is essentially based on the use of set-valued mappings of a special kind, called fans, for local approximation. Convex sets of linear operators provide an example of fans. Generally, fans can be considered a natural set-valued extension of linear operators. The first part of the paper presents a study of fans; the second is devoted to calculus and includes extensions of the main theorems of classical calculus.Introduction. The idea to extend the framework of differential calculus so as to cover more general classes of functions and mappings is by no means new. Basically, it was the underlying idea for the differentiation theory connected with the Lebesgue integral and for the theory of distributions. Both theories deal essentially with what could be called nonlocal aspects of the calculus centered around the Newton-Leibniz and integration by parts formulae. The notion of the value of a derivative at a given point makes no sense in either of them.Nonsmooth analysis appeared in the 1970's just to carry out an extension of the local aspect of the calculus connected with the idea of (linear) approximation of a mapping about a given point. Certain separate ideas and results appeared of course much earlier (one could recall the Dini numbers for instance) but a systematic study began during the last decade when the natural development of the optimization theory made the need for such an extension very acute and, as often happens, practical and heuristic computations were initiated before an adequate theory appeared (see [52] and references therein).It is not surprising that the main impulse came from the optimization theory which has natural mechanisms generating nonsmoothness. But as a result, most of the efforts were applied to obtain more and more refined conditions for extrema with less interest in those aspects of analysis that are less immediately connected with this purpose. The only exception was perhaps the generalized gradients of Clarke whose analytical virtues were recognized from the beginning ([8]-[12], [17],

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confidence: 99%

Nonsmooth analysis: differential calculus of nondifferentiable mappings

Ioffe¹

1981

Trans. Amer. Math. Soc.

205

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“…It has been shown [5,27] that the fulfillment of MFCQ is not enough for guaranteeing that local minimizers of nonlinear programming problems satisfy SONC. In [3] it has been proved that the condition (MFCQ and WCR) (4) is a suitable constraint qualification that guarantees that local minimizers satisfy SONC.…”

Section: Application To An Augmented Lagrangian Methodsmentioning

confidence: 99%

Second-order negative-curvature methods for box-constrained and general constrained optimization

et al. 2009

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A Nonlinear Programming algorithm that converges to second-order stationary points is introduced in this paper. The main tool is a second-order negative-curvature method for box-constrained minimization of a certain class of functions that do not possess continuous second derivatives. This method is used to define an Augmented Lagrangian algorithm of PHR (Powell-Hestenes-Rockafellar) type. Convergence proofs under weak constraint qualifications are given. Numerical examples showing that the new method converges to second-order stationary points in situations in which first-order methods fail are exhibited.

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“…The extension of condition (1 1) to problems with vector-valued performance index was developed by GorokhovikZ0* 2 2 (see also Reference 25). Condition (14) and its direct proof may be found in the work of G o r~k h o v i k .~~. 34 The classical extremal uo(t), t E T , is said to be normal for problem (1)- (5) if the cone of Lagrange vectors L(uo) is a ray, i.e.…”

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confidence: 99%

High‐order necessary optimality conditions for control problems with terminal constraints

Gorokhovik¹

1983

Optim Control Appl Methods

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SUMMARYThis is a survey of high-order necessary optimality conditions for control problems with inequality and equality terminal constraints. Both classical and Pontryagin extremals are considered. For classical extremals, the characteristics of necessary optimality conditions in terms of variations are given, and the investigations devoted to Kelley's conditions are discussed in detail. For Pontryagin extremals, two types of results are given: generalizations and extensions of matrix impulse optimality condition and high-order maximum principle.

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Conditions of High Order for a Local Minimum in Problems With Constraints

Abstract: A novel drive circuit, useful for medical electronics, is capable of supplying a sample of human tissue, across which there should be zero direct voltage (dc), with a well-defined test current from a source having an output impedance exceeding 16 M at 100 kHz.

Cited by 95 publications

References 26 publications

Nonsmooth analysis: differential calculus of nondifferentiable mappings

Nonsmooth analysis: differential calculus of nondifferentiable mappings

Second-order negative-curvature methods for box-constrained and general constrained optimization

High‐order necessary optimality conditions for control problems with terminal constraints

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