Optimality conditions for robust weakly efficient solutions in uncertain optimization

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  • Published: 13 February 2024

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  • Yuwen Zhai 1 ,
  • Qilin Wang   ORCID: orcid.org/0000-0001-7968-051X 1 ,
  • Tian Tang 2 &
  • Maoyuan Lv 1  

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In this paper, we find the flimsily robust weakly efficient solution to the uncertain vector optimization problem by means of the weighted sum scalarization method and strictly robust counterpart. In addition, we introduce a higher-order weak upper inner Studniarski epiderivative of set-valued maps, and obtain two properties of the new notion under the assumption of the star-shaped set. Finally, by applying the higher-order weak upper inner Studniarski epiderivative, we obtain a sufficient and necessary optimality condition of the vector-based robust weakly efficient solution to an uncertain vector optimization problem under the condition of the higher-order strictly generalized cone convexity. As applications, the corresponding optimality conditions of the robust (weakly) Pareto solutions are obtained by the current methods.

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Acknowledgements

This research was partially supported by the National Natural Science Foundation of China (No.11971078), the Group Building Project for Scientific Innovation for Universities in Chongqing (CXQT21021) and the Natural Natural Science Foundation of Chongqing (CSTB2023NSCQ-MSX1071).

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Zhai, Y., Wang, Q., Tang, T. et al. Optimality conditions for robust weakly efficient solutions in uncertain optimization. Optim Lett (2024). https://doi.org/10.1007/s11590-023-02085-7

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Robust solutions of optimization problems affected by uncertain probabilities

  • Aharon Ben-Tal
  • Dick den Hertog
  • Anja De Waegenaere
  • Bertrand Melenberg
  • Gijs Rennen

In this paper we focus on robust linear optimization problems with uncertainty regions defined by phi-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on phi-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector. Such problems frequently occur in, for example, optimization problems in inventory control or finance that involve terms containing moments of random variables, expected utility, etc. We show that the robust counterpart of a linear optimization problem with phi-divergence uncertainty is tractable for most of the choices of phi typically considered in the literature. We extend the results to problems that are nonlinear in the optimization variables. Several applications, including an asset pricing example and a numerical multi-item newsvendor example, illustrate the relevance of the proposed approach.

CentER Discussion Paper CDP 2011-061, May 2011, CentER, Department of Econometrics and Operations Research, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands

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Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

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2013, Management Science

In this paper we focus on robust linear optimization problems with uncertainty regions defined by ϕ-divergences (for example, chi-squared, Hellinger, Kullback–Leibler). We show how uncertainty regions based on ϕ-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector. Such problems frequently occur in, for example, optimization problems in inventory control or finance that involve terms containing moments of random variables, expected utility, etc. We show that the robust counterpart of a linear optimization problem with ϕ-divergence uncertainty is tractable for most of the choices of ϕ typically considered in the literature. We extend the results to problems that are nonlinear in the optimization variables. Several applications, including an asset pricing example and a numerical multi-item newsvendor example, illustrate the relevance of the proposed approach. This paper was accepted by Gérard P. Cachon, optimization.

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Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

Ben-Tal, A. , den Hertog, D. , De Waegenaere, A.M.B. , Melenberg, B. , Rennen, G.

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In this paper we focus on robust linear optimization problems with uncertainty regions defined by [phi]-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on [phi]-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector. Such problems frequently occur in, for example, optimization problems in inventory control or finance that involve terms containing moments of random variables, expected utility, etc. We show that the robust counterpart of a linear optimization problem with [phi]-divergence uncertainty is tractable for most of the choices of [phi] typically considered in the literature. We extend the results to problems that are nonlinear in the optimization variables. Several applications, including an asset pricing example and a numerical multi-item newsvendor example, illustrate the relevance of the proposed approach. This paper was accepted by Gérard P. Cachon, optimization.(This abstract was borrowed from another version of this item.)

robust solutions of optimization problems affected by uncertain probabilities

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T1 - Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

AU - Ben-Tal, A.

AU - den Hertog, D.

AU - De Waegenaere, A.M.B.

AU - Melenberg, B.

AU - Rennen, G.

N2 - In this paper we focus on robust linear optimization problems with uncertainty regions defined by ø-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on ø-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector. Such problems frequently occur in, for example, optimization problems in inventory control or finance that involve terms containing moments of random variables, expected utility, etc. We show that the robust counterpart of a linear optimization problem with ø-divergence uncertainty is tractable for most of the choices of ø typically considered in the literature. We extend the results to problems that are nonlinear in the optimization variables. Several applications, including an asset pricing example and a numerical multi-item newsvendor example, illustrate the relevance of the proposed approach.

AB - In this paper we focus on robust linear optimization problems with uncertainty regions defined by ø-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on ø-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector. Such problems frequently occur in, for example, optimization problems in inventory control or finance that involve terms containing moments of random variables, expected utility, etc. We show that the robust counterpart of a linear optimization problem with ø-divergence uncertainty is tractable for most of the choices of ø typically considered in the literature. We extend the results to problems that are nonlinear in the optimization variables. Several applications, including an asset pricing example and a numerical multi-item newsvendor example, illustrate the relevance of the proposed approach.

KW - robust optimization

KW - ø-divergence

KW - goodness-of-fit statistics

M3 - Discussion paper

VL - 2011-061

T3 - CentER Discussion Paper

BT - Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

PB - Operations research

CY - Tilburg

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    In this paper we focus on robust linear optimization problems with uncertainty regions defined by ϕ -divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on ϕ -divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector.

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    Ben-Tal et al.: Robust Solutions of Optimization Problems Affected by Uncertain Probabilities Management Science 59(2), pp. 341-357, ©2013 INFORMS (p-divergence functionals, the resulting robust coun terpart problem is polynomially solvable. In fact, in many cases it reduces to a linear, or a conic quadratic problem.

  3. PDF Robust Solutions of Optimization Problems Affected by Uncertain

    Robust solutions of optimization problems affected by uncertain probabilities AharonBen-Tal∗ Department of Industrial Engineering and Management, Technion - Israel Institute of Technology, Haifa 32000, Israel CentER Extramural Fellow, CentER, Tilburg University, The Netherlands DickdenHertog,AnjaDeWaegenaere,BertrandMelenberg,GijsRennen

  4. Robust Solutions of Optimization Problems Affected by Uncertain ...

    In this paper we focus on robust linear optimization problems with uncertainty regions defined by ø-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on ø-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector.

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    In this paper we focus on robust linear optimization problems with uncertainty regions defined by φ-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on Φ-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector.

  6. Robust Solutions of Optimization Problems Affected by Uncertain

    Abstract: In this paper we focus on robust linear optimization problems with uncertainty regions defined by [phi]-divergences (for example, chi-squared, Hellinger, Kullback-Leibler).

  7. Robust Solutions of Optimization Problems Affected by Uncertain

    The robust counterpart of a linear optimization problem with φ-divergence uncertainty is tractable for most of the choices of φ typically considered in the literature and extended to problems that are nonlinear in the optimization variables. Expand View on SSRN research.tilburguniversity.edu Save to Library Create Alert Cite Topics AI-Generated

  8. Robust Solutions of Optimization Problems Affected by Uncertain

    In this paper we focus on robust linear optimization problems with uncertainty regions defined by ø-divergences (for example, chi-squared, Hellinger, Kullback-L ... Copy URL. Copy DOI. Robust Solutions of Optimization Problems Affected by Uncertain Probabilities. CentER Working Paper Series No. 2011-061. 29 ... 50140 and 608059, 608059, Robust ...

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    Robust Solutions of Optimization Problems Affected by Uncertain Probabilities Authors: Aharon Ben-Tal , Dick den Hertog , Anja De Waegenaere , Bertrand Melenberg , Gijs Rennen Authors Info & Claims Management Science Volume 59 Issue 2 02 2013 pp 341-357 https://doi.org/10.1287/mnsc.1120.1641 Published: 01 February 2013 Publication History 164 0

  10. Robust Solutions of Optimization Problems Affected by Uncertain

    In this paper we focus on robust linear optimization problems with uncertainty regions defined by ϕ-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regi...

  11. Optimality conditions for robust weakly efficient solutions in

    Uncertainty is a common challenge in optimization problems, which has led to the development of uncertain optimization approaches including stochastic optimization and robust optimization [].While stochastic optimization relies on mathematical expectations or probabilities, robust optimization assumes that uncertain parameters belong to a certain set, and aims to find a solution which is ...

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    In this paper we focus on robust linear optimization problems with uncertainty regions defined by ϕ-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how...

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    Aharon Ben-Tal & Dick den Hertog & Anja De Waegenaere & Bertrand Melenberg & Gijs Rennen, 2013. "Robust Solutions of Optimization Problems Affected by Uncertain Probabilities," Management Science, INFORMS, vol. 59(2), pages 341-357, April.

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    In this paper, we propose a methodology for constructing uncertainty sets within the framework of robust optimization for linear optimization problems with uncertain parameters. Our approach relies on decision maker risk preferences. Specifically, we utilize the theory of coherent risk measures initiated by Artzner et al. (1999) [Artzner, P., F ...

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    In this paper we focus on robust linear optimization problems with uncertainty regions defined by phi-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on phi-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector.

  16. Robust Solutions of Optimization Problems Affected by Uncertain ...

    In this paper we focus on robust linear optimization problems with uncertainty regions defined by ø-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on ø-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector.

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    The idea of robust optimization is to define a so-called uncertainty region for the uncertain parameters, and then to require that the constraint should hold for all parameter values in this uncertainty region. The optimization problem modeling this requirement is called the Robust Counterpart Problem (RCP).

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    This paper shows how uncertainty regions based on φ-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector and extends the results to problems that are nonlinear in the optimization variables. Expand optimization-online.org Save to Library Create Alert Cite

  19. Robust Solutions of Optimization Problems Affected by Uncertain

    In this paper we focus on robust linear optimization problems with uncertainty regions defined by [phi]-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on [phi]-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector.

  20. Robust Solutions of Optimization Problems Affected by Uncertain

    In this paper we focus on robust linear optimization problems with uncertainty regions defined by ø-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on ø-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector.

  21. [PDF] Globalized Robust Optimization for Nonlinear Uncertain

    A globalized robust counterpart of the classical robust counterpart (RC) of the problem requires the solution to be feasible for all uncertain parameter values in a so-called uncertainty set, and offers no guarantees for parameter values outside this uncertainty set. Robust optimization is a methodology that can be applied to problems that are affected by uncertainty in the problem's parameters.

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    In this paper we focus on robust linear optimization problems with uncertainty regions defined by [phi]-divergences (for example, chi-squared, Hellinger, Kullback-Leibler).

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    In this paper, we find the flimsily robust weakly efficient solution to the uncertain vector optimization problem by means of the weighted sum scalarization method and strictly robust counterpart.

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    Samenvatting In this paper we focus on robust linear optimization problems with uncertainty regions defined by ø-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on ø-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector.

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    In this paper we focus on robust linear optimization problems with uncertainty regions defined by [phi]-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on [phi]-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector.