American Institute of Mathematical Sciences

Advanced
April  2007, 3(2): 209-222. doi: 10.3934/jimo.2007.3.209

A smoothing scheme for optimization problems with Max-Min constraints

X. X. Huang 1, , Xiaoqi Yang 2, and  K. L. Teo 3,

1. 

School of Management, Fudan University, Shanghai 200433
2. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong
3. 

Department of Mathematics and Statistics, Curtin University of Technology, Perth

Received  August 2006 Revised  January 2007 Published  April 2007

In this paper, we apply a smoothing approach to a minimization problem with a max-min constraint (i.e., a min-max-min problem). More specifically, we first rewrite the min-max-min problem as an optimization problem with several min-constraints and then approximate each min-constraint function by a smooth function. As a result, the original min-max-min optimization problem can be solved by solving a sequence of smooth optimization problems. We investigate the relationship between the global optimal value and optimal solutions of the original min-max-min optimization problem and that of the approximate smooth problem. Under some conditions, we show that the limit points of the first-order (second-order) stationary points of the smooth optimization problems are first-order (second-order) stationary points of the original min-max-min optimization problem.
Keywords: convergence, smooth approximation, Min-max-min problem, optimality condition..
Mathematics Subject Classification: Primary: 90C30; Secondary: 90C46, 90C4.
Citation: X. X. Huang, Xiaoqi Yang, K. L. Teo. A smoothing scheme for optimization problems with Max-Min constraints. Journal of Industrial & Management Optimization, 2007, 3 (2) : 209-222. doi: 10.3934/jimo.2007.3.209
[1]

Jinchuan Zhou, Changyu Wang, Naihua Xiu, Soonyi Wu. First-order optimality conditions for convex semi-infinite min-max programming with noncompact sets. Journal of Industrial & Management Optimization, 2009, 5 (4) : 851-866. doi: 10.3934/jimo.2009.5.851
[2]

Roberto Livrea, Salvatore A. Marano. A min-max principle for non-differentiable functions with a weak compactness condition. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1019-1029. doi: 10.3934/cpaa.2009.8.1019
[3]

José M. Amigó, Ángel Giménez. Formulas for the topological entropy of multimodal maps based on min-max symbols. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3415-3434. doi: 10.3934/dcdsb.2015.20.3415
[4]

Baolan Yuan, Wanjun Zhang, Yubo Yuan. A Max-Min clustering method for $k$-means algorithm of data clustering. Journal of Industrial & Management Optimization, 2012, 8 (3) : 565-575. doi: 10.3934/jimo.2012.8.565
[5]

Meixia Li, Changyu Wang, Biao Qu. Non-convex semi-infinite min-max optimization with noncompact sets. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1859-1881. doi: 10.3934/jimo.2017022
[6]

Gaoxi Li, Zhongping Wan, Jia-wei Chen, Xiaoke Zhao. Necessary optimality condition for trilevel optimization problem. Journal of Industrial & Management Optimization, 2020, 16 (1) : 55-70. doi: 10.3934/jimo.2018140
[7]

Jianxin Zhou. Optimization with some uncontrollable variables: a min-equilibrium approach. Journal of Industrial & Management Optimization, 2007, 3 (1) : 129-138. doi: 10.3934/jimo.2007.3.129
[8]

R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497
[9]

Xianjin Chen, Jianxin Zhou. A local min-orthogonal method for multiple solutions of strongly coupled elliptic systems. Conference Publications, 2009, 2009 (Special) : 151-160. doi: 10.3934/proc.2009.2009.151
[10]

G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279
[11]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[12]

Blaine Keetch, Yves van Gennip. A Max-Cut approximation using a graph based MBO scheme. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6091-6139. doi: 10.3934/dcdsb.2019132
[13]

Shaojun Lan, Yinghui Tang, Miaomiao Yu. System capacity optimization design and optimal threshold $N^{*}$ for a $GEO/G/1$ discrete-time queue with single server vacation and under the control of Min($N, V$)-policy. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1435-1464. doi: 10.3934/jimo.2016.12.1435
[14]

Nazih Abderrazzak Gadhi, Khadija Hamdaoui. Optimality results for a specific fractional problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020157
[15]

Wenxing Zhu, Yanpo Liu, Geng Lin. Speeding up a memetic algorithm for the max-bisection problem. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 151-168. doi: 10.3934/naco.2015.5.151
[16]

Dieudonné Nijimbere, Songzheng Zhao, Xunhao Gu, Moses Olabhele Esangbedo, Nyiribakwe Dominique. Tabu search guided by reinforcement learning for the max-mean dispersion problem. Journal of Industrial & Management Optimization, 2021, 17 (6) : 3223-3246. doi: 10.3934/jimo.2020115
[17]

A. C. Eberhard, C.E.M. Pearce. A sufficient optimality condition for nonregular problems via a nonlinear Lagrangian. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 301-331. doi: 10.3934/naco.2012.2.301
[18]

Ana P. Lemos-Paião, Cristiana J. Silva, Delfim F. M. Torres. A sufficient optimality condition for delayed state-linear optimal control problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2293-2313. doi: 10.3934/dcdsb.2019096
[19]

Lucian Coroianu, Danilo Costarelli, Sorin G. Gal, Gianluca Vinti. Approximation by multivariate max-product Kantorovich-type operators and learning rates of least-squares regularized regression. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4213-4225. doi: 10.3934/cpaa.2020189
[20]

Marvin S. Müller. Approximation of the interface condition for stochastic Stefan-type problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4317-4339. doi: 10.3934/dcdsb.2019121

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (38)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences