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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  May 2020 Early access  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 and Management Optimization, 2020, 16 (3) : 1221-1233. doi: 10.3934/jimo.2018201
References:
[1]

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

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[13]

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

[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. 

[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.

[16]

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

[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. 

[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. 

[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.

[20]

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

[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. 

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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. 

show all references

References:
[1]

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

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[13]

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

[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. 

[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.

[16]

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

[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. 

[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. 

[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.

[20]

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

[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. 

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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. 

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