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On the entropic regularization method for solving min-max problems with applications

Author

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  • Xing-Si Li
  • Shu-Cherng Fang

Abstract

Consider a min-max problem in the form of min xεX max 1≤i≤m {f i (x)}. It is well-known that the non-differentiability of the max functionF(x) ≡ max 1≤i≤m {f i (x)} presents difficulty in finding an optimal solution. An entropic regularization procedure provides a smooth approximationF p (x) that uniformly converges toF(x) overX with a difference bounded by ln(m)/p, forp > 0. In this way, withp being sufficiently large, minimizing the smooth functionF p (x) overX provides a very accurate solution to the min-max problem. The same procedure can be applied to solve systems of inequalities, linear programming problems, and constrained min-max problems. Copyright Physica-Verlag 1997

Suggested Citation

  • Xing-Si Li & Shu-Cherng Fang, 1997. "On the entropic regularization method for solving min-max problems with applications," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 46(1), pages 119-130, February.
    • Handle: RePEc:spr:mathme:v:46:y:1997:i:1:p:119-130
    DOI: 10.1007/BF01199466
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    Citations

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    Cited by:

    1. Fusheng Wang & Kecun Zhang, 2008. "A hybrid algorithm for nonlinear minimax problems," Annals of Operations Research, Springer, vol. 164(1), pages 167-191, November.
    2. Helene Krieg & Tobias Seidel & Jan Schwientek & Karl-Heinz Küfer, 2022. "Solving continuous set covering problems by means of semi-infinite optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(1), pages 39-82, August.
    3. Birbil, S.I. & Fang, S-C. & Han, J., 2002. "Entropic Regularization Approach for Mathematical Programs with Equilibrium Constraints," ERIM Report Series Research in Management ERS-2002-71-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    4. Birbil, S.I. & Fang, S-C. & Han, J., 2002. "Entropic regularization approach for mathematical programs with equilibrium constraints," Econometric Institute Research Papers EI 2002-52, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    5. R. L. Sheu & S. Y. Wu, 1999. "Combined Entropic Regularization and Path-Following Method for Solving Finite Convex Min-max Problems Subject to Infinitely Many Linear Constraints," Journal of Optimization Theory and Applications, Springer, vol. 101(1), pages 167-190, April.
    6. Alidaee, Bahram, 2014. "Zero duality gap in surrogate constraint optimization: A concise review of models," European Journal of Operational Research, Elsevier, vol. 232(2), pages 241-248.
    7. Xide Zhu & Peijun Guo, 2017. "Approaches to four types of bilevel programming problems with nonconvex nonsmooth lower level programs and their applications to newsvendor problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(2), pages 255-275, October.

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