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Comparison Between Quasidifferentials and Exhausters

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  • Majid E. Abbasov

    (St. Petersburg State University, SPbSU)

Abstract

Directional derivatives play one of the major roles in optimization. Optimality conditions can be described in terms of these objects. These conditions, however, are not constructive. To overcome this problem, one has to represent the directional derivative in special forms. Two such forms are quasidifferentials and exhausters proposed by V.F. Demyanov. Quasidifferentials were introduced in 1980s. Optimality conditions in terms of these objects were developed by L.N. Polyakova and V.F. Demyanov. It was described how to find directions of steepest descent and ascent when these conditions are not satisfied. This paved a way for constructing new optimization algorithms. Quasidifferentials allow one to treat a wide class of functions. V.F. Demyanov introduced the notion of exhausters in 2000s to expand the class of functions that can be treated. It should be noted that a great contribution to the emergence of this notion was made by B.N. Pshenichny and A.M. Rubinov. In this work it is shown that exhausters not only allow one to treat a wider class of functions than quasidifferentials (since every quasidifferentiable function has exhausters) but is also preferable even for quasidifferentiable functions when solving nonsmooth optimization problems.

Suggested Citation

  • Majid E. Abbasov, 2017. "Comparison Between Quasidifferentials and Exhausters," Journal of Optimization Theory and Applications, Springer, vol. 175(1), pages 59-75, October.
    • Handle: RePEc:spr:joptap:v:175:y:2017:i:1:d:10.1007_s10957-017-1167-3
    DOI: 10.1007/s10957-017-1167-3
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    References listed on IDEAS

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    1. Mahide Küçük & Ryszard Urbański & Jerzy Grzybowski & Yalçın Küçük & İlknur Atasever Güvenç & Didem Tozkan & Mustafa Soyertem, 2015. "Reduction of Weak Exhausters and Optimality Conditions via Reduced Weak Exhausters," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 693-707, June.
    2. Elena K. Basaeva & Anatoly G. Kusraev & Semen S. Kutateladze, 2016. "Quaisidifferentials in Kantorovich Spaces," Journal of Optimization Theory and Applications, Springer, vol. 171(2), pages 365-383, November.
    3. M. Abbasov & V. Demyanov, 2013. "Proper and adjoint exhausters in nonsmooth analysis: optimality conditions," Journal of Global Optimization, Springer, vol. 56(2), pages 569-585, June.
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    Cited by:

    1. Majid E. Abbasov, 2020. "Optimality conditions for an exhausterable function on an exhausterable set," Journal of Global Optimization, Springer, vol. 76(1), pages 57-67, January.
    2. Didem Tozkan, 2022. "On reduction of exhausters via a support function representation," Journal of Global Optimization, Springer, vol. 82(1), pages 105-118, January.
    3. Kabgani, Alireza & Soleimani-damaneh, Majid, 2022. "Semi-quasidifferentiability in nonsmooth nonconvex multiobjective optimization," European Journal of Operational Research, Elsevier, vol. 299(1), pages 35-45.
    4. Alireza Kabgani, 2021. "Characterization of Nonsmooth Quasiconvex Functions and their Greenberg–Pierskalla’s Subdifferentials Using Semi-Quasidifferentiability notion," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 666-678, May.
    5. Vivek Laha & Harsh Narayan Singh, 2023. "On quasidifferentiable mathematical programs with equilibrium constraints," Computational Management Science, Springer, vol. 20(1), pages 1-20, December.

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