A NEW SMOOTHING FUNCTION TECHNIQUE FOR SOLVING MINIMAX PROBLEMS

Nurullah Yilmaz

doi:10.11948/20240361

2025 Volume 15 Issue 3

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Nurullah Yilmaz. A NEW SMOOTHING FUNCTION TECHNIQUE FOR SOLVING MINIMAX PROBLEMS[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1703-1718. doi: 10.11948/20240361

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Nurullah Yilmaz. A NEW SMOOTHING FUNCTION TECHNIQUE FOR SOLVING MINIMAX PROBLEMS[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1703-1718. doi: 10.11948/20240361

A NEW SMOOTHING FUNCTION TECHNIQUE FOR SOLVING MINIMAX PROBLEMS

Nurullah Yilmaz^1, ,

1.
Department of Mathematics, Suleyman Demirel University, Cunur Campus, 32100 Isparta, Türkiye

Corresponding author: Email: nurullahyilmaz@sdu.edu.tr(N. Yilmaz)

Abstract

Abstract

In this study, we consider non-smooth finite minimax problems. A new approach for solving minimax problems is developed, employing indicator functions and smoothing functions. First, the formulation of minimax problems is revised using indicator functions. Then, a new generation smoothing technique is used for the revised formulation. An algorithm is developed to solve the revised and smoothed problems numerically. The efficiency of the algorithm is demonstrated on several test problems, and a comparison is conducted between the numerical results achieved and those of alternative approaches. Finally, the portfolio planning problem is considered as a real-life application, and satisfactory results are obtained.
- Minimax problem /
- non-smooth optimization /
- smoothing technique
MSC: 90C30, 65K05, 65D10

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References

[1]	T. Antczak, The minimax exact penalty fuzzy function method for solving convex nonsmooth optimization problems with fuzzy objective functions, J. Ind. Manag. Optim., 2024, 20(1), 392–427. doi: 10.3934/jimo.2023083 CrossRef Google Scholar
[2]	E. M. Arkin, R. Hassin and A. Levin, Approximations for minimum and min-max vehicle routing problems, J. Algorithm., 2006, 59(1), 1–18. doi: 10.1016/j.jalgor.2005.01.007 CrossRef Google Scholar
[3]	A. Bagirov, A. A. Nuaimat and N. Sultanova, Hyperbolic smoothing function method for minimax problems, Optimization, 2013, 62(6), 759–782. doi: 10.1080/02331934.2012.675335 CrossRef Google Scholar
[4]	A. M. Bagirov, N. Sultanova, A. Al Nuaimat and S. Taheri, Solving minimax problems: Local smoothing versus global smoothing, in Numerical Analysis and Optimization (Edited by M. Al-Baali, L. Grandinetti and A. Purnama), Springer International Publishing, Cham, 2018, 23–43. Google Scholar
[5]	D. P. Bertsekas, On penalty and multiplier methods for constrained minimization, SIAM J. Control Optim., 1976, 14(2), 216–235. doi: 10.1137/0314017 CrossRef Google Scholar
[6]	X. Cai, K. -L. Teo, X. Yang and X. Y. Zhou, Portfolio optimization under a minimax rule, Manag. Sci., 2000, 46(7), 957–972. Google Scholar
[7]	C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Comput. Optim. Appl., 1996, 5(2), 97–138. doi: 10.1007/BF00249052 CrossRef Google Scholar
[8]	X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Programm., 2012, 134(1), 71–99. doi: 10.1007/s10107-012-0569-0 CrossRef Google Scholar
[9]	V. F. Demyanov and V. N. Malozemov, Introduction Minimax, Wiley, New York, 1974. Google Scholar
[10]	L. Dong and B. Yu, A spline smoothing newton method for finite minimax problems, J. Eng. Math., 2015, 93(1), 145–158. doi: 10.1007/s10665-014-9733-2 CrossRef Google Scholar
[11]	D. Z. Du and P. M. Pardalos, Minimax and Applications, 4, Kluwer Academic, Dordrecht, 1995. Google Scholar
[12]	J. Du, L. Zhao, J. Feng and X. Chu, Computation offloading and resource allocation in mixed fog/cloud computing systems with min-max fairness guarantee, IEEE Trans. Commun., 2018, 66(4), 1594–1608. doi: 10.1109/TCOMM.2017.2787700 CrossRef Google Scholar
[13]	T. Ergenc, S. W. Pickl, N. Radde and G. W. Weber, Generalized semi-infinite optimization and anticipatory systems, International Journal of Computing Anticipatory Systems, 2004, 15, 3–30. Google Scholar
[14]	W. Fan, J. Liang, H. C. So and G. Lu, Min-max metric for spectrally compatible waveform design via log-exponential smoothing, IEEE Trans. Signal Process., 2020, 68, 1075–1090. doi: 10.1109/TSP.2020.2969043 CrossRef Google Scholar
[15]	Y. Feng, L. Hongwei, Z. Shuisheng and L. Sanyang, A smoothing trust-region newton-cg method for minimax problem, Appl. Math. Comput., 2008, 199(2), 581–589. Google Scholar
[16]	A. Fuduli, M. Gaudioso, G. Giallombardo and G. Miglionico, A partially inexact bundle method for convex semi-infinite minmax problems, Commun. Nonlinear Sci. Numer. Simul., 2015, 21(1), 172–180. Numerical Computations: Theory and Algorithms (NUMTA 2013), International Conference and Summer School. Google Scholar
[17]	M. Gaudioso, G. Giallombardo and G. Miglionico, An incremental method for solving convex finite min-max problems, Math. Oper. Res., 2006, 31(1), 173–187. doi: 10.1287/moor.1050.0175 CrossRef Google Scholar
[18]	C. Grossman, Smoothing techniques for exact penalty function methods, American Mathematical Society, Providence, Rhode Island, 2016, 249–265. Google Scholar
[19]	Z. Cobandag Guloglu and G. W. Weber, Risk modeling in optimization problems via value at risk, conditional value at risk, and its robustification, in Modeling, Dynamics, Optimization and Bioeconomics Ⅱ (Edited by A. A. Pinto and D. Zilberman), Springer International Publishing, Cham., 2017, 133–145. Google Scholar
[20]	H. T. Jongen and O. Stein, Smoothing by mollifiers, part i: Semi-infinite optimization, J. Global Optim., 2008, 41, 319–334. doi: 10.1007/s10898-007-9232-3 CrossRef Google Scholar
[21]	H. T. Jongen and O. Stein, Smoothing by mollifiers, part ii: Nonlinear optimization, J. Global Optim., 2008, 41, 335–350. doi: 10.1007/s10898-007-9231-4 CrossRef Google Scholar
[22]	H. T. Jongen, F. Twilt and G. W. Weber, Semi-infinite optimization: Structure and stability of the feasible set, J. Optim. Theory Appl., 1992, 72, 529–552. doi: 10.1007/BF00939841 CrossRef Google Scholar
[23]	J. Liu and L. Zheng, A smoothing iterative method for the finite minimax problem, J. Comput. Appl. Math., 2020, 374, 112741. doi: 10.1016/j.cam.2020.112741 CrossRef Google Scholar
[24]	L. Luksan and J. Vlcek, Test problems for non-smooth unconstrained and linearly constrained optimization, Technical Reports No. 798, Institute of Computer Science, Academy of Sciences of the Czech Republic, 2000. Google Scholar
[25]	S. Mahmoudinazlou and C. Kwon, A hybrid genetic algorithm for the min–max multiple traveling salesman problem, Comput. Oper. Res., 2024, 162, 106455. doi: 10.1016/j.cor.2023.106455 CrossRef Google Scholar
[26]	A. Nezami and M. Mortezaee, A new gradient-based neural dynamic framework for solving constrained min-max optimization problems with an application in portfolio selection model, Appl. Intell., 2019, 49(2), 369–419. Google Scholar
[27]	A. Özmen, E. Kropat and G. -W. Weber, Robust optimization in spline regression models for multi-model regulatory networks under polyhedral uncertainty, Optimization, 2017, 66(12), 2135–2155. doi: 10.1080/02331934.2016.1209672 CrossRef Google Scholar
[28]	B. Akteke-Öztürk, G. -W. Weber and G. Köksal, Optimization of generalized desirability functions under model uncertainty, Optimization, 2017, 66(12), 2157–2169. doi: 10.1080/02331934.2017.1371167 CrossRef Google Scholar
[29]	B. Akteke-Öztürk, G. -W. Weber and G. Köksal, Generalized desirability functions: A structural and topological analysis of desirability functions, Optimization, 2020. Google Scholar
[30]	C. Papahristodoulou and E. Dotzauer, Optimal portfolios using linear programming models, J. Oper. Res. Soc., 2004, 55(11), 1169–1177. doi: 10.1057/palgrave.jors.2601765 CrossRef Google Scholar
[31]	E. Polak, On the mathematical foundations of nondifferentiable optimization in engineering design, SIAM Review, 1987, 29(1), 21–89. doi: 10.1137/1029002 CrossRef Google Scholar
[32]	E. Polak, J. O. Royset and R. S. Womersley, Algorithms with adaptive smoothing for finite minimax problems, J. Optim. Theory Appl., 2003, 119(3), 459–484. doi: 10.1023/B:JOTA.0000006685.60019.3e CrossRef Google Scholar
[33]	A. Sahiner, G. Kapusuz and N. Yilmaz, A new smoothing approach to exact penalty functions for inequality constrained optimization problems, Numer. Algebra Control Optim., 2016, 6(2), 161–173. doi: 10.3934/naco.2016006 CrossRef Google Scholar
[34]	W. Sánchez, C. Arias and R. Perez, A jacobian smoothing inexact newton method for solving the nonlinear complementary problem, Comput. Appl. Math., 2024, 43(5), 279. doi: 10.1007/s40314-024-02775-7 CrossRef Google Scholar
[35]	M. Souza, A. E. Xavier, C. Lavor and N. Maculan, Hyperbolic smoothing and penalty techniques applied to molecular structure determination, Oper. Res. Lett., 2011, 39(6), 461–465. doi: 10.1016/j.orl.2011.07.007 CrossRef Google Scholar
[36]	K. L. Teo and X. Yang, Portfolio selection problem with minimax type risk function, Ann. Oper. Res., 2001, 101, 333–349. doi: 10.1023/A:1010909632198 CrossRef Google Scholar
[37]	F. Guerra Vazquez, H. Günzel and H. Jongen, On logarithmic smoothing of the maximum function, Ann. Oper. Res., 2001, 101(1), 209–220. Google Scholar
[38]	H. Wang, A decentralized smoothing quadratic regularization algorithm for composite consensus optimization with non-lipschitz singularities, Numer. Algorithm., 2024, 96(1), 369–396. doi: 10.1007/s11075-023-01650-6 CrossRef Google Scholar
[39]	A. E. Xavier, The hyperbolic smoothing clustering method, Pattern Recogn., 2010, 43(3), 731–737. doi: 10.1016/j.patcog.2009.06.018 CrossRef Google Scholar
[40]	S. Xu, Smoothing method for minimax problems, Comput. Optim. Appl., 2001, 20(3), 267–279. doi: 10.1023/A:1011211101714 CrossRef Google Scholar
[41]	N. Yilmaz and A. Kayacan, A new smoothing algorithm to solve a system of nonlinear inequalities, Fundamental Journal of Mathematics and Applications, 2023, 6(3), 137–146. doi: 10.33401/fujma.1261409 CrossRef Google Scholar
[42]	N. Yilmaz and A. Sahiner, New smoothing approximations to piecewise smooth functions and applications, Numer. Funct. Anal. Optim., 2019, 40(5), 513–534. doi: 10.1080/01630563.2018.1561466 CrossRef Google Scholar
[43]	N. Yilmaz and A. Sahiner, Smoothing techniques in solving non-lipschitz absolute value equations, Int. J. Comput. Math., 2023, 100(4), 867–879. doi: 10.1080/00207160.2022.2163388 CrossRef Google Scholar
[44]	H. Yin, An adaptive smoothing method for continuous minimax problems, Asia-Pac. J. Oper. Res., 2015, 32(01), 1540001. doi: 10.1142/S0217595915400011 CrossRef Google Scholar
[45]	I. Zang, A smoothing-out technique for min–max optimization, Math. Programm., 1980, 19(1), 61–77. Google Scholar
[46]	Z. Zhou and Y. Duan, An aggregate homotopy method for solving unconstrained minimax problems, Optimization, 2021, 70(8), 1791–1808. Google Scholar
[47]	Z. Zhou and Q. Yang, An active set smoothing method for solving unconstrained minimax problems, Math. Probl. Eng., 2020, 2020, 9108150. Google Scholar