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May  2020, 16(3): 1221-1233. doi: 10.3934/jimo.2018201

Stability analysis for generalized semi-infinite optimization problems under functional perturbations

Department of Mathematics, Bohai University, Jinzhou, Liaoning 121013, China

* Corresponding author: Xiaodong Fan (E-mail address: bhdxfxd@163.com)

Received  June 2017 Revised  October 2017 Published  December 2018

Fund Project: The first author is supported by National Natural Science Foundation of China (No. 61572082), Natural Science Foundation of Liaoning Province of China (No. 20170540004, 20170540012) and Educational Commission of Liaoning Province of China (No. LZ2016003)

The concepts of essential solutions and essential solution sets for generalized semi-infinite optimization problems (GSIO for brevity) are introduced under functional perturbations, and the relations among the concepts of essential solutions, essential solution sets and lower semicontinuity of solution mappings are discussed. We show that a solution is essential if and only if the solution is unique; and a solution subset is essential if and only if it is the solution set itself. Some sufficient conditions for the upper semicontinuity of solution mappings are obtained. Finally, we show that every GSIO problem can be arbitrarily approximated by stable GSIO problems (the solution mapping is continuous), i.e., the set of all stable GSIO problems is dense in the set of all GSIO problems with the given topology.

Citation: Xiaodong Fan, Tian Qin. Stability analysis for generalized semi-infinite optimization problems under functional perturbations. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1221-1233. doi: 10.3934/jimo.2018201
References:
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J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Wiley, Chichester, 1998. Google Scholar

[2]

M. J. CánovasA. L. DontchevM. A. López and J. Parra, Metric regularity of semiinfinite constraint systems, Math. Program., 104 (2005), 329-346.  doi: 10.1007/s10107-005-0618-z.  Google Scholar

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M. J. CánovasD. KlatteM. A. López and J. Parra, Metric regularity in convex semi-infinite optimization under canonical perturbations, SIAM. J. Optim., 18 (2007), 717-732.  doi: 10.1137/060658345.  Google Scholar

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M. J. CánovasM. A. LópezB. S. Mordukhovich and J. Parra, Variational analysis in semi-Infinite and infinite programming, Ⅰ: stability of linear inequality systems of feasible solutions, SIAM J. Optim., 20 (2009), 1504-1526.  doi: 10.1137/090765948.  Google Scholar

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M. J. CánovasM. A. LópezB. S. Mordukhovich and J. Parra, Variational Analysis in Semi-Infinite and Infinite Programming, Ⅱ: Necessary Optimality Conditions, SIAM J. Optim., 20 (2010), 2788-2806.  doi: 10.1137/09076595X.  Google Scholar

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G. Y. Chen and B. D. Craven, Existence and continuity of solutions for vector optimization, J. Optim. Theory Appl., 81 (1994), 459-468.  doi: 10.1007/BF02193095.  Google Scholar

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T. D. ChuongN. Q. Huy and J. C. Yao, Stability of semi-infinite vector optimization problems under functional perturbations, J. Glob. Optim., 45 (2009), 583-595.  doi: 10.1007/s10898-008-9391-x.  Google Scholar

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X. FanC. Cheng and H. Wang, Essential solutions of parametric vector optimization problems, Pacific J. of Optimization, 9 (2013), 413-425.   Google Scholar

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X. FanC. Cheng and H. Wang, Stability of semi-infinite vector optimization problems without compact constraints, Nonlinear Anal., 74 (2011), 2087-2093.  doi: 10.1016/j.na.2010.11.013.  Google Scholar

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X. FanC. Cheng and H. Wang, Stability analysis for vector quasiequilibrium problems, Positivity, 17 (2013), 365-379.  doi: 10.1007/s11117-012-0172-x.  Google Scholar

[11]

X. FanC. Cheng and H. Wang, Sensitivity analysis for vector equilibrium problems under functional perturbations, Numer. Funct. Anal. Optim., 35 (2014), 564-575.  doi: 10.1080/01630563.2013.814140.  Google Scholar

[12]

X. FanC. Cheng and H. Wang, Density of stable convex semi-infinite vector optimization problems, Oper. Res. Lett., 40 (2012), 140-143.  doi: 10.1016/j.orl.2011.11.010.  Google Scholar

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M. K. Fort, Essential and nonessential fixed points, Amer. J. Math., 72 (1950), 315-322.  doi: 10.2307/2372035.  Google Scholar

[14]

A. FuC. Dong and L. Wang, An experimental study on stability and generalization of extreme learning machines, Int. J. Mach. Learn. Cyb., 6 (2015), 129-135.   Google Scholar

[15]

M. A. GobernaM. A. López and M. Todorov, Stability theory for linear inequality systems. Ⅱ. Upper semicontinuity of the solution set mapping, SIAM J. Optim., 7 (1997), 1138-1151.  doi: 10.1137/S105262349528901X.  Google Scholar

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S. Kinoshita, On essential component of the set of fixed points, Osaka J. Math., 4 (1952), 19-22.   Google Scholar

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Z. LinH. Yang and J. Yu, On existence and essential components of the solution set for the system of vector quasi-equilibrium problems, Nonlinear Anal., 63 (2005), e2445-e2452.   Google Scholar

[18]

D. Liu and Y. Du, New results of stability analysis for a class of neutral-type neural network with mixed time delays, Int. J. Mach. Learn. Cyb., 6 (2015), 555-566.   Google Scholar

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Q. Luo, Essential component and essential optimum solution of optimization problems, J. Optim. Theory Appl., 102 (1999), 433-438.  doi: 10.1023/A:1021740709876.  Google Scholar

[20]

J. R. Munkres, Topology, 2nd edition, Prentice Hall, New Jersey, 2000.  Google Scholar

[21]

D. T. Peng, Essential solutions and essential components of the solution set of infinite-dimensional vector optimization problems, Math. Appl., 22 (2009), 358-364.   Google Scholar

[22]

S. W. Xiang and W. S. Yin, Stability results for efficient solutions of vector optimization problems, J. Optim. Theory Appl., 134 (2007), 385-398.  doi: 10.1007/s10957-007-9214-0.  Google Scholar

[23]

S. W. Xiang and Y. H. Zhou, Continuity properties of solutions of vector optimization, Nonlinear Anal., 64 (2006), 2496-2506.  doi: 10.1016/j.na.2005.08.029.  Google Scholar

[24]

S. W. Xiang and Y. H. Zhou, On essential sets and essential components of efficient solutions for vector optimization problems, J. Math. Anal. Appl., 315 (2006), 317-326.  doi: 10.1016/j.jmaa.2005.06.077.  Google Scholar

[25]

H. Yang and J. Yu, Essential solutions and essential components of solution set of vector quasi-equilibrium problems, J. Systems Sci. Math. Sci., 24 (2004), 74-84.   Google Scholar

[26]

J. Yu, Essential weak efficient solution in multiobjective optimization problems, J. Math. Anal. Appl., 166 (1992), 230-235.  doi: 10.1016/0022-247X(92)90338-E.  Google Scholar

[27]

J. Yu, Essential equilibria of n-person noncooperative games, J. Math. Econ., 31 (1999), 361-372.  doi: 10.1016/S0304-4068(97)00060-8.  Google Scholar

[28]

J. Yu and S. W. Xiang, On essential component of the set of Nash equilibrium points, Nonlinear Anal., 38 (1999), 259-264.  doi: 10.1016/S0362-546X(98)00193-X.  Google Scholar

[29]

X. ZhangR. LiC. Han and R. Yao, Robust stability analysis of uncertain genetic regulatory networks with mixed time delays, Int. J. Mach. Learn. Cyb., 7 (2016), 1005-1022.   Google Scholar

show all references

References:
[1]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Wiley, Chichester, 1998. Google Scholar

[2]

M. J. CánovasA. L. DontchevM. A. López and J. Parra, Metric regularity of semiinfinite constraint systems, Math. Program., 104 (2005), 329-346.  doi: 10.1007/s10107-005-0618-z.  Google Scholar

[3]

M. J. CánovasD. KlatteM. A. López and J. Parra, Metric regularity in convex semi-infinite optimization under canonical perturbations, SIAM. J. Optim., 18 (2007), 717-732.  doi: 10.1137/060658345.  Google Scholar

[4]

M. J. CánovasM. A. LópezB. S. Mordukhovich and J. Parra, Variational analysis in semi-Infinite and infinite programming, Ⅰ: stability of linear inequality systems of feasible solutions, SIAM J. Optim., 20 (2009), 1504-1526.  doi: 10.1137/090765948.  Google Scholar

[5]

M. J. CánovasM. A. LópezB. S. Mordukhovich and J. Parra, Variational Analysis in Semi-Infinite and Infinite Programming, Ⅱ: Necessary Optimality Conditions, SIAM J. Optim., 20 (2010), 2788-2806.  doi: 10.1137/09076595X.  Google Scholar

[6]

G. Y. Chen and B. D. Craven, Existence and continuity of solutions for vector optimization, J. Optim. Theory Appl., 81 (1994), 459-468.  doi: 10.1007/BF02193095.  Google Scholar

[7]

T. D. ChuongN. Q. Huy and J. C. Yao, Stability of semi-infinite vector optimization problems under functional perturbations, J. Glob. Optim., 45 (2009), 583-595.  doi: 10.1007/s10898-008-9391-x.  Google Scholar

[8]

X. FanC. Cheng and H. Wang, Essential solutions of parametric vector optimization problems, Pacific J. of Optimization, 9 (2013), 413-425.   Google Scholar

[9]

X. FanC. Cheng and H. Wang, Stability of semi-infinite vector optimization problems without compact constraints, Nonlinear Anal., 74 (2011), 2087-2093.  doi: 10.1016/j.na.2010.11.013.  Google Scholar

[10]

X. FanC. Cheng and H. Wang, Stability analysis for vector quasiequilibrium problems, Positivity, 17 (2013), 365-379.  doi: 10.1007/s11117-012-0172-x.  Google Scholar

[11]

X. FanC. Cheng and H. Wang, Sensitivity analysis for vector equilibrium problems under functional perturbations, Numer. Funct. Anal. Optim., 35 (2014), 564-575.  doi: 10.1080/01630563.2013.814140.  Google Scholar

[12]

X. FanC. Cheng and H. Wang, Density of stable convex semi-infinite vector optimization problems, Oper. Res. Lett., 40 (2012), 140-143.  doi: 10.1016/j.orl.2011.11.010.  Google Scholar

[13]

M. K. Fort, Essential and nonessential fixed points, Amer. J. Math., 72 (1950), 315-322.  doi: 10.2307/2372035.  Google Scholar

[14]

A. FuC. Dong and L. Wang, An experimental study on stability and generalization of extreme learning machines, Int. J. Mach. Learn. Cyb., 6 (2015), 129-135.   Google Scholar

[15]

M. A. GobernaM. A. López and M. Todorov, Stability theory for linear inequality systems. Ⅱ. Upper semicontinuity of the solution set mapping, SIAM J. Optim., 7 (1997), 1138-1151.  doi: 10.1137/S105262349528901X.  Google Scholar

[16]

S. Kinoshita, On essential component of the set of fixed points, Osaka J. Math., 4 (1952), 19-22.   Google Scholar

[17]

Z. LinH. Yang and J. Yu, On existence and essential components of the solution set for the system of vector quasi-equilibrium problems, Nonlinear Anal., 63 (2005), e2445-e2452.   Google Scholar

[18]

D. Liu and Y. Du, New results of stability analysis for a class of neutral-type neural network with mixed time delays, Int. J. Mach. Learn. Cyb., 6 (2015), 555-566.   Google Scholar

[19]

Q. Luo, Essential component and essential optimum solution of optimization problems, J. Optim. Theory Appl., 102 (1999), 433-438.  doi: 10.1023/A:1021740709876.  Google Scholar

[20]

J. R. Munkres, Topology, 2nd edition, Prentice Hall, New Jersey, 2000.  Google Scholar

[21]

D. T. Peng, Essential solutions and essential components of the solution set of infinite-dimensional vector optimization problems, Math. Appl., 22 (2009), 358-364.   Google Scholar

[22]

S. W. Xiang and W. S. Yin, Stability results for efficient solutions of vector optimization problems, J. Optim. Theory Appl., 134 (2007), 385-398.  doi: 10.1007/s10957-007-9214-0.  Google Scholar

[23]

S. W. Xiang and Y. H. Zhou, Continuity properties of solutions of vector optimization, Nonlinear Anal., 64 (2006), 2496-2506.  doi: 10.1016/j.na.2005.08.029.  Google Scholar

[24]

S. W. Xiang and Y. H. Zhou, On essential sets and essential components of efficient solutions for vector optimization problems, J. Math. Anal. Appl., 315 (2006), 317-326.  doi: 10.1016/j.jmaa.2005.06.077.  Google Scholar

[25]

H. Yang and J. Yu, Essential solutions and essential components of solution set of vector quasi-equilibrium problems, J. Systems Sci. Math. Sci., 24 (2004), 74-84.   Google Scholar

[26]

J. Yu, Essential weak efficient solution in multiobjective optimization problems, J. Math. Anal. Appl., 166 (1992), 230-235.  doi: 10.1016/0022-247X(92)90338-E.  Google Scholar

[27]

J. Yu, Essential equilibria of n-person noncooperative games, J. Math. Econ., 31 (1999), 361-372.  doi: 10.1016/S0304-4068(97)00060-8.  Google Scholar

[28]

J. Yu and S. W. Xiang, On essential component of the set of Nash equilibrium points, Nonlinear Anal., 38 (1999), 259-264.  doi: 10.1016/S0362-546X(98)00193-X.  Google Scholar

[29]

X. ZhangR. LiC. Han and R. Yao, Robust stability analysis of uncertain genetic regulatory networks with mixed time delays, Int. J. Mach. Learn. Cyb., 7 (2016), 1005-1022.   Google Scholar

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