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C.c. Scaief - One of the best experts on this subject based on the ideXlab platform.

Mustafa Soyertem - One of the best experts on this subject based on the ideXlab platform.

  • reduction of Exhausters by set order relations and cones
    arXiv: Optimization and Control, 2021
    Co-Authors: Mustafa Soyertem, İlknur Atasever Güvenç, Didem Tozkan
    Abstract:

    The notions of upper and lower Exhausters are effective tools for the study of non smooth functions. There are many studies presenting optimality conditions for unconstrained and constrained cases. One can observe that optimality conditions in terms of both proper and adjoint Exhausters are related to all elements of the Exhausters. Moreover, in the constrained case the conditions that must be provided for a particular cone determined by constraint set and the point (to be checked whether it is optimal) are rather challenging to check. Thus it is advantageous to reduce the number of sets in the exhauster for constrained case. In this work, we first consider constrained optimization problems and deal with the problem of reducing generalized Exhausters of the directional derivative of the objective function. We present some results to reduce generalized lower (upper) Exhausters by using set order relations $\preceq^{m_1}$ and $\preceq^{m_2}$, respectively. Furthermore, we show that a generalized exhauster $E$ can be reduced to the set of minimal elements of $E$ with respect to $\preceq^{m_1}$ or $\preceq^{m_2}$. Then considering unconstrained optimization problems, lower Exhausters are reduced by using cones.

  • some relationships among quasidifferential weak subdifferential and Exhausters
    Optimization, 2016
    Co-Authors: Mahide Küçük, Jerzy Grzybowski, Yalçın Küçük, Ryszard Urbański, İlknur Atasever Güvenç, Didem Tozkan, Mustafa Soyertem
    Abstract:

    In this study, some relationships between quasidifferentiability and weakly subdifferentiability of a function are investigated, and some results on calculating the quasidifferential in terms of weak Exhausters are presented. Moreover, under some assumptions weak subdifferentiability of a quasidifferentiable function is proved and the weak subdifferential is obtained by using the quasidifferential. Under some assumptions, it is proved that a positively homogeneous and lower semicontinuous function is quasidifferentiable. It is also shown that directional differentiability and weak subdifferentiability imply quasidifferentiability. Furthermore, a method to evaluate the quasidifferential is given by using reduced weak lower Exhausters.

  • weak subdifferential superdifferential weak Exhausters and optimality conditions
    Optimization, 2015
    Co-Authors: Mahide Küçük, Jerzy Grzybowski, Yalçın Küçük, Ryszard Urbański, Didem Tozkan, Atasever I Guvenc, Mustafa Soyertem
    Abstract:

    In this work, the notion of weak superdifferential is presented. Some calculation rules are given to evaluate weak subdifferential and weak superdifferential of some classes of functions represented by support functions. Moreover, some methods are obtained to calculate weak subdifferential of convex functions. In addition, the concept of weak lower and weak upper Exhausters of positively homogeneous functions are introduced by using weak subdifferential and weak superdifferential, respectively. In terms of weak Exhausters, some optimality conditions are given to find local or global minimizers/maximizers of some classes of functions.

  • Reduction of Weak Exhausters and Optimality Conditions via Reduced Weak Exhausters
    Journal of Optimization Theory and Applications, 2015
    Co-Authors: Mahide Küçük, Jerzy Grzybowski, Yalçın Küçük, Ryszard Urbański, İlknur Atasever Güvenç, Didem Tozkan, Mustafa Soyertem
    Abstract:

    In this work, it is proved that weak Exhausters of a positively homogeneous function can be reduced if weak subdifferential/superdifferential can be represented as the sum of a subset of it and weak subdifferential of zero function. It is also shown that weak Exhausters can be reduced if this subset is the set of minimal elements of weak subdifferential/superdifferential with respect to weak subdifferential of zero function. At the end, some optimality conditions are given using reduced weak Exhausters.

Soyertem Mustafa - One of the best experts on this subject based on the ideXlab platform.

  • Reduction of Exhausters by Set Order Relations and Cones
    2021
    Co-Authors: Soyertem Mustafa, Güvenç, İlknur Atasever, Tozkan Didem
    Abstract:

    The notions of upper and lower Exhausters are effective tools for the study of non smooth functions. There are many studies presenting optimality conditions for unconstrained and constrained cases. One can observe that optimality conditions in terms of both proper and adjoint Exhausters are related to all elements of the Exhausters. Moreover, in the constrained case the conditions that must be provided for a particular cone determined by constraint set and the point (to be checked whether it is optimal) are rather challenging to check. Thus it is advantageous to reduce the number of sets in the exhauster for constrained case. In this work, we first consider constrained optimization problems and deal with the problem of reducing generalized Exhausters of the directional derivative of the objective function. We present some results to reduce generalized lower (upper) Exhausters by using set order relations $\preceq^{m_1}$ and $\preceq^{m_2}$, respectively. Furthermore, we show that a generalized exhauster $E$ can be reduced to the set of minimal elements of $E$ with respect to $\preceq^{m_1}$ or $\preceq^{m_2}$. Then considering unconstrained optimization problems, lower Exhausters are reduced by using cones.Comment: 17 pages, 2 figure

  • Reduction of Weak Exhausters and Optimality Conditions via Reduced Weak Exhausters
    'Springer Science and Business Media LLC', 2015
    Co-Authors: Küçük M., Urba?ski R., Grzybowski J., Küçük Y., Güvenç İ.a., Tozkan D., Soyertem Mustafa
    Abstract:

    In this work, it is proved that weak Exhausters of a positively homogeneous function can be reduced if weak subdifferential/superdifferential can be represented as the sum of a subset of it and weak subdifferential of zero function. It is also shown that weak Exhausters can be reduced if this subset is the set of minimal elements of weak subdifferential/superdifferential with respect to weak subdifferential of zero function. At the end, some optimality conditions are given using reduced weak Exhausters. © 2014, Springer Science+Business Media New York

  • Weak subdifferential/superdifferential, weak Exhausters and optimality conditions
    'Informa UK Limited', 2015
    Co-Authors: Küçük M., Urba?ski R., Grzybowski J., Küçük Y., Tozkan D., Atasever Güvenç İ., Soyertem Mustafa
    Abstract:

    In this work, the notion of weak superdifferential is presented. Some calculation rules are given to evaluate weak subdifferential and weak superdifferential of some classes of functions represented by support functions. Moreover, some methods are obtained to calculate weak subdifferential of convex functions. In addition, the concept of weak lower and weak upper Exhausters of positively homogeneous functions are introduced by using weak subdifferential and weak superdifferential, respectively. In terms of weak Exhausters, some optimality conditions are given to find local or global minimizers/maximizers of some classes of functions. © 2014 Taylor & Francis.Firat University Scientific Research Projects Management Unit: 1201F018This study was supported by Anadolu University Scientific Research Projects Commission [grant number 1201F018]

Mahide Küçük - One of the best experts on this subject based on the ideXlab platform.

  • some relationships among quasidifferential weak subdifferential and Exhausters
    Optimization, 2016
    Co-Authors: Mahide Küçük, Jerzy Grzybowski, Yalçın Küçük, Ryszard Urbański, İlknur Atasever Güvenç, Didem Tozkan, Mustafa Soyertem
    Abstract:

    In this study, some relationships between quasidifferentiability and weakly subdifferentiability of a function are investigated, and some results on calculating the quasidifferential in terms of weak Exhausters are presented. Moreover, under some assumptions weak subdifferentiability of a quasidifferentiable function is proved and the weak subdifferential is obtained by using the quasidifferential. Under some assumptions, it is proved that a positively homogeneous and lower semicontinuous function is quasidifferentiable. It is also shown that directional differentiability and weak subdifferentiability imply quasidifferentiability. Furthermore, a method to evaluate the quasidifferential is given by using reduced weak lower Exhausters.

  • weak subdifferential superdifferential weak Exhausters and optimality conditions
    Optimization, 2015
    Co-Authors: Mahide Küçük, Jerzy Grzybowski, Yalçın Küçük, Ryszard Urbański, Didem Tozkan, Atasever I Guvenc, Mustafa Soyertem
    Abstract:

    In this work, the notion of weak superdifferential is presented. Some calculation rules are given to evaluate weak subdifferential and weak superdifferential of some classes of functions represented by support functions. Moreover, some methods are obtained to calculate weak subdifferential of convex functions. In addition, the concept of weak lower and weak upper Exhausters of positively homogeneous functions are introduced by using weak subdifferential and weak superdifferential, respectively. In terms of weak Exhausters, some optimality conditions are given to find local or global minimizers/maximizers of some classes of functions.

  • Reduction of Weak Exhausters and Optimality Conditions via Reduced Weak Exhausters
    Journal of Optimization Theory and Applications, 2015
    Co-Authors: Mahide Küçük, Jerzy Grzybowski, Yalçın Küçük, Ryszard Urbański, İlknur Atasever Güvenç, Didem Tozkan, Mustafa Soyertem
    Abstract:

    In this work, it is proved that weak Exhausters of a positively homogeneous function can be reduced if weak subdifferential/superdifferential can be represented as the sum of a subset of it and weak subdifferential of zero function. It is also shown that weak Exhausters can be reduced if this subset is the set of minimal elements of weak subdifferential/superdifferential with respect to weak subdifferential of zero function. At the end, some optimality conditions are given using reduced weak Exhausters.

O D Nelson - One of the best experts on this subject based on the ideXlab platform.