On Necessary Conditions for a Minimum

Ioffe, A. D.

doi:10.1007/s10958-016-3003-y

Cited by 5 publications

(3 citation statements)

References 24 publications

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“…One can verify that the pair (x * , u * ) ≡ (0, 0) satisfies various existing optimality conditions for this problem in terms of subdifferentials and normal cones [11,12,17,19,47,51,55,66,[70][71][72]81]. We leave the laborious task of verifying these conditions to the interested reader.…”

Section: Discussionmentioning

confidence: 99%

“…It seems that since the mid-90s nonsmooth problems of the calculus of variations became just an auxiliary tool for the derivation of optimality conditions for nonsmooth optimal control problems and nonsmooth variational problems involving differential inclusions (cf. [47,48,51]). As a result, relatively little attention has been paid to nonsmooth multidimensional problems of the calculus of variations, as well as problems with additional constraints, such as nonsmooth isoperimetric problems and problems with additional constraints at the boundary.…”

Section: Introductionmentioning

confidence: 99%

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Constrained nonsmooth problems of the calculus of variations

Долгополик

2021

ESAIM: COCV

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The paper is devoted to an analysis of optimality conditions for nonsmooth multidimensional problems of the calculus of variations with various types of constraints, such as additional constraints at the boundary and isoperimetric constraints. To derive optimality conditions, we study generalised concepts of differentiability of nonsmooth functions called codifferentiability and quasidifferentiability. Under some natural and easily verifiable assumptions we prove that a nonsmooth integral functional defined on the Sobolev space is continuously codifferentiable and compute its codifferential and quasidifferential. Then we apply general optimality conditions for nonsmooth optimisation problems in Banach spaces to obtain optimality conditions for nonsmooth problems of the calculus of variations. Through a series of simple examples we demonstrate that our optimality conditions are sometimes better than existing ones in terms of various subdifferentials, in the sense that our optimality conditions can detect the non-optimality of a given point, when subdifferential-based optimality conditions fail to disqualify this point as non-optimal.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Constrained nonsmooth problems of the calculus of variations

Долгополик

2021

ESAIM: COCV

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“…Analysis of such a problem needs different techniques and we refer to [52,101], where necessary optimality conditions for the problem were obtained along these lines. A more general result was established a few years later by Clarke [25] (actually the most general for optimal control of differential inclusions so far) but a shorter proof of Clarke's theorem based on optimality alternative is now also available [60].…”

Section: Optimal Control Of Differential Inclusionmentioning

confidence: 98%

Metric Regularity—a Survey Part ii. applications

Ioffe¹

2016

J. Aust. Math. Soc.

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Metric regularity theory lies in the very heart of variational analysis, a relatively new discipline whose appearance was, to a large extent, determined by the needs of modern optimization theory in which such phenomena as nondifferentiability and set-valued mappings naturally appear. The roots of the theory go back to such fundamental results of the classical analysis as the implicit function theorem, Sard theorem and some others. The paper offers a survey of the state of the art of some principal parts of the theory along with a variety of its applications in analysis and optimization.

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Search for Locally Optimal Strategies in a Linear Game Problem with Favorable Situations

Mamatov

2024

J. Appl. Ind. Math.

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On Necessary Conditions for a Minimum

Cited by 5 publications

References 24 publications

Constrained nonsmooth problems of the calculus of variations

Constrained nonsmooth problems of the calculus of variations

Metric Regularity—a Survey Part ii. applications

Search for Locally Optimal Strategies in a Linear Game Problem with Favorable Situations

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