On global unconstrained minimization of the difference of polyhedral functions

Polyakova, Lyudmila N.

doi:10.1007/s10898-010-9589-6

Cited by 10 publications

(10 citation statements)

References 13 publications

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“…Therefore, in the minimization problems, it is essential that the approximation is bounded from below. Polyakova derived this condition in terms of codifferential in [28].…”

Section: Boundedness Conditionsmentioning

confidence: 99%

“…Then we give definitions of codifferentials and coexhausters and connect these notions with the class of difference of polyhedral convex functions. In Section 2 we present Polyakova's (see [28]) boundedness condition in terms of codifferentials. We prove that this condition is equivalent to the condition of boundedness stated in [29] in terms of coexhausters.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Therefore, in the minimization problems, it is essential that the approximation is bounded from below. Polyakova derived this condition in terms of codifferential in [28].…”

Section: Boundedness Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Directional differentiability, coexhausters, codifferentials and polyhedral DC functions

Abbasov¹

2021

Preprint

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“…Using these notations, by (5) we see that the subdifferentials of the convex functions g, h, defined in (1) and (2), are as follows…”

Section: Remark 21 the Inclusion "⊃" In (8) Was Proved By Roshchina mentioning

confidence: 99%

“…While the condition is necessary for a point to be a local minimizer of a general d.c. function (see, e.g., [4, Subsection 2.2], [5,Section 5]), its sufficiency is proved only for the case where the second d.c. component is polyhedral, see, e.g., [4]. Necessity of this condition can be considered as a consequence of optimality condition obtained by Demy'anov et al in quasidifferential calculus; see, for instance, [6, Theorem 5.1] and [7, Theorem 3.1].…”

Section: Optimality Conditionsmentioning

confidence: 99%

“…functions can be treated as special problems in d.c. (difference of convex functions) programming and, therefore, quasidifferentiable programming. One may derive properties of such problems from general theory of d.c. programming (see, e.g., [4,5]) and of quasidifferential calculus (see the books by Demy'anov and Vasil'ev [6], and by Demy'anov and Rubinov [7,8]). Some properties of the unconstrained model were considered by Polyakova [5] using conjugate functions and hypodifferentials of polyhedral functions.…”

Section: Introductionmentioning

confidence: 99%

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On the Problem of Minimizing a Difference of Polyhedral Convex Functions Under Linear Constraints

Hang

Yen

2015

J Optim Theory Appl

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This paper is concerned with two d.p. (difference of polyhedral convex functions) programming models, unconstrained and linearly constrained, in a finitedimensional setting. We obtain exact formulae for the Fréchet and Mordukhovich subdifferentials of a d.p. function. We establish optimality conditions via subdifferentials in the sense of convex analysis, of Fréchet and of Mordukhovich, and describe their relationships. Existence and computation of descent and steepest descent directions for both the models are also studied.

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