Minimization of a quasi-differentiable function in a quasi-differentiable set

Demyanov, V. F.; Polyakova, Lyudmila N.

doi:10.1016/0041-5553(80)90269-4

Cited by 17 publications

(6 citation statements)

References 1 publication

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“…The known results for necessary and sufficient optimality conditions for QD functions by [16], [19, Chap. V, Theorem 3.1] carry over to the directed subdifferential.…”

Section: Optimality Conditions Descent and Ascent Directionsmentioning

confidence: 99%

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The directed and Rubinov subdifferentials of quasidifferentiable functions, Part II: Calculus

Baier

Farkhi

Roshchina

2012

Nonlinear Analysis: Theory, Methods & Applications

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International audienceWe continue the study of the directed subdifferential for quasidifferentiable functions started in [R. Baier, E. Farkhi, V. Roshchina, The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I: Definition and examples (this journal)]. Calculus rules for the directed subdifferentials of sum, product, quotient, maximum and minimum of quasidifferentiable functions are derived. The relation between the Rubinov subdifferential and the subdifferentials of Clarke, Dini, Michel-Penot, and Mordukhovich is discussed. Important properties implying the claims of Ioffe's axioms as well as necessary and sufficient optimality conditions for the directed subdifferential are obtained

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“…The known results for necessary and sufficient optimality conditions for QD functions by [16], [19, Chap. V, Theorem 3.1] carry over to the directed subdifferential.…”

Section: Optimality Conditions Descent and Ascent Directionsmentioning

confidence: 99%

“…We now state sufficient optimality conditions for the directed subdifferential (see [16], [19,Secs. V.1 and V.3] and [20]). Proposition 4.4.…”

Section: Optimality Conditions Descent and Ascent Directionsmentioning

confidence: 99%

The directed and Rubinov subdifferentials of quasidifferentiable functions, Part II: Calculus

Baier

Farkhi

Roshchina

2012

Nonlinear Analysis: Theory, Methods & Applications

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“…One important advantage of this approach is that all the tools and methods can be built and used not only theoretically but also in practical problems. The approach goes back to the early 80-th when Demyanov, Rubinov and Polyakova proposed and studied the notion of quasidifferentials [2][3][4][5]. Quasidifferentials are pairs of convex compact sets that enable one to represent the directional derivative of a function at a point in a form of sum of maximum and minimum of a linear functions.…”

Section: Introductionmentioning

confidence: 99%

Directional differentiability, coexhausters, codifferentials and polyhedral DC functions

Abbasov¹

2021

Preprint

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Codifferentials and coexhausters are used to describe nonhomogeneous approximations of a nonsmooth function. Despite the fact that coexhausters are modern generalizations of codifferentials, the theories of these two concepts continue to develop simultaneously. Moreover, codifferentials and coexhausters are strongly connected with DC functions. In this paper we trace analogies between all these objects, and prove the equivalence of the boundedness and optimality conditions described in terms of these notions. This allows one to extend the results derived in terms of one object to the problems stated via the other one. Another contribution of this paper is the study of connection between nonhomogeneous approximations and directional derivatives and formulate optimality conditions in terms of nonhomogeneous approximations.

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“…Necessary and sufficient conditions for an unconstrained local minimum in terms of quasidifferentials were first obtained in [10,54]. In [9], Demyanov and Polyakova studied optimality conditions in terms of quasidifferentials for the problem min f 0 (x) subject to h(x) ≤ 0.…”

Section: Introductionmentioning

confidence: 99%

“…[46, Example 1]), optimality conditions for quasidifferentiable programming problems cannot be formulated in the traditional way involving the Lagrangian function, which results in the fact that optimality conditions for such problems can be stated in several non-equivalent forms. Optimality conditions for problem (1) from [9] were formulated in geometric terms and involved some cones generated by a quasidifferential of the constraint. Optimality conditions for problem (1) similar to Fritz John and KKT conditions in which Lagrange multipliers depend on individual elements of quasidifferentials were studied in [43,46,59].…”

Section: Introductionmentioning

confidence: 99%

A New Constraint Qualification and Sharp Optimality Conditions for Nonsmooth Mathematical Programming Problems in Terms of Quasidifferentials

Dolgopolik

2019

Preprint

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This paper is devoted to an analysis of optimality conditions for nonsmooth mathematical programming problems with equality and inequality constraints in terms of Demyanov-Rubinov-Polyakova quasidifferentials. To this end, we obtain a novel description of convex subcones of the contingent cone to a set defined by quasidifferentiable equality and inequality constraints. With the use of this description we derive optimality conditions for nonsmooth mathematical programming problems in terms of quasidifferentials based on a new constraint qualification. The main feature of these optimality conditions and constraint qualification is the fact that they depend on individual elements of quasidifferentials of constraints. To illustrate the theoretical results, we present two simple examples in which the optimality conditions obtained in this paper are not satisfied at a given point, while optimality conditions in terms of various subdifferentials (in fact, any outer semicontinuous/limiting subdifferential) fail to disqualify this point as nonoptimal.

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Minimization of a quasi-differentiable function in a quasi-differentiable set

Cited by 17 publications

References 1 publication

The directed and Rubinov subdifferentials of quasidifferentiable functions, Part II: Calculus

The directed and Rubinov subdifferentials of quasidifferentiable functions, Part II: Calculus

Directional differentiability, coexhausters, codifferentials and polyhedral DC functions

A New Constraint Qualification and Sharp Optimality Conditions for Nonsmooth Mathematical Programming Problems in Terms of Quasidifferentials

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