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Second-order composed contingent derivatives of perturbation maps in set-valued optimization

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Abstract

In the paper, we study calculus rules of second-order composed contingent derivatives. More precisely, chain rule and sum rule are established and their applications to some particular mathematical models are obtained. Then sensitivity analysis in set-valued optimization using second-order composed contingent derivatives are proposed. Our results are new and many examples are given to illustrate them.

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References

  • Anh NLH (2017a) Sensitivity analysis in constrained set-valued optimization via Studniarski derivatives. Positivity 21:255–272

    Article  MathSciNet  Google Scholar 

  • Anh NLH (2017b) Some results on sensitivity analysis in set-valued optimization. Positivity 21:1527–1543

    Article  MathSciNet  Google Scholar 

  • Anh NLH, Khanh PQ (2013) Higher-order optimality conditions in set-valued optimization using radial sets and radial derivatives. J Glob Optim 56:519–536

    Article  MathSciNet  Google Scholar 

  • Anh NLH, Khanh PQ (2014) Higher-order optimality conditions for proper efficiency in nonsmooth vector optimization using radial sets and radial derivatives. J Glob Optim 58:693–709

    Article  MathSciNet  Google Scholar 

  • Anh NLH, Khanh PQ, Tung LT (2011) Higher-order radial derivatives and optimality conditions in nonsmooth vector optimization. Nonlinear Anal TMA 74:7365–7379

    Article  MathSciNet  Google Scholar 

  • Aubin JP, Frankowska H (1990) Set-valued analysis. Birkhauser, Boston

    MATH  Google Scholar 

  • Bigi G, Castellani M (2002) \(K\)-epiderivatives for set-valued functions and optimization. Math Methods Oper Res 55:401–412

    Article  MathSciNet  Google Scholar 

  • Chen GY, Jahn J (1998) Optimality conditions for set-valued optimization problems. Math Methods Oper Res 48:187–200

    Article  MathSciNet  Google Scholar 

  • Chen CR, Li SJ, Teo KL (2009) Solution semicontinuity of parametric generalized vector equilibirum problems. J Glob Optim 45:309–318

    Article  Google Scholar 

  • Chuong TD (2011) Clarke coderivatives of efficient point multifunctions in parametric vector optimization. Nonlinear Anal 74:273–285

    Article  MathSciNet  Google Scholar 

  • Chuong TD (2013) Derivatives of the efficient point multifunction in parametric vector optimization problems. J Optim Theory Appl 156:247–265

    Article  MathSciNet  Google Scholar 

  • Crepsi GP, Ginchev I, Rocca M (2006) First order optimality conditions in set-valued optimization. Math Methods Oper Res 63:87–106

    Article  MathSciNet  Google Scholar 

  • Durea M (2008) Optimality conditions for weak and firm efficiency in set-valued optimization. J Math Anal Appl 344:1018–1028

    Article  MathSciNet  Google Scholar 

  • Fiacco AV (1983) Introduction to sensitivity and stability analysis in nonlinear programming. Academic Press, New York

    MATH  Google Scholar 

  • Gong XH (2008) Continuity of the solution set to parametric weak vector equilibrium problems. J Optim Theory Appl 139:35–46

    Article  MathSciNet  Google Scholar 

  • Huy NQ, Mordukhovich BS, Yao J-C (2008) Coderivatives of frontier and solution maps in parametric multiobjective optimization. Taiwan J Math 12:2083–2111

    Article  MathSciNet  Google Scholar 

  • Khan AA, Isac G (2009) Second-order optimality conditions in set-valued optimization by a new tangential derivative. Acta Math Vietnam 34:81–90

    MathSciNet  MATH  Google Scholar 

  • Khan AA, Ward DE (2012) Toward second-order sensitivity analysis in set-valued optimization. J Nonlinear Convex Anal 13:65–83

    MathSciNet  MATH  Google Scholar 

  • Khan AA, Tammer C, Zǎlinescu C (2015) Set-valued optimization: an introduction with applications. Springer, Heidelberg

    Book  Google Scholar 

  • Kimura K, Yao JC (2008) Semicontinuity of solution mappings of parametric generalized vector equilibrium problems. J Optim Theory Appl 138:429–443

    Article  MathSciNet  Google Scholar 

  • Kuk H, Tanino T, Tanaka M (1996) Sensitivity analysis in vector optimization. J Optim Theory Appl 89:713–730

    Article  MathSciNet  Google Scholar 

  • Lee GM, Huy NQ (2006) On proto-differentiability of generalized perturbation maps. J Math Anal Appl 324:1297–1309

    Article  MathSciNet  Google Scholar 

  • Levy AB, Mordukhovich BS (2004) Coderivatives in parametric optimization. Math Program Ser A 99:311–327

    Article  MathSciNet  Google Scholar 

  • Li SJ, Li MH (2011) Sensitivity analysis of parametric weak vector equilibrium problems. J Math Anal Appl 380:354–362

    Article  MathSciNet  Google Scholar 

  • Li SJ, Teo KL, Yang XQ (2008) Higher-order Mond-Weir duality for set-valued optimization. J Comput Appl Math 217:339–349

    Article  MathSciNet  Google Scholar 

  • Li SJ, Meng KW, Penot JP (2009) Calculus rules for derivatives of multimaps. Set-Valued Anal 17:21–39

    Article  MathSciNet  Google Scholar 

  • Mordukhovich BS (2006a) Variational analysis and generalized differentiation, vol I: basic theory. Springer, Berlin

  • Mordukhovich BS (2006b) Variational analysis and generalized differentiation, vol II: applications. Springer, Berlin

  • Mordukhovich BS (2018) Variational analysis and applications. Springer, Berlin

    Book  Google Scholar 

  • Penot JP (1984) Differentiability of relations and differential stability of perturbed optimization problems. SIAM J Control Optim 22:529–551

    Article  MathSciNet  Google Scholar 

  • Robinson SM (1976) Stability theory for systems of inequalities, part II. Differentiable nonlinear systems. SIAM J Numer Anal 13:497–513

    Article  MathSciNet  Google Scholar 

  • Rockafellar RT (1989) Proto-differentiability of set-valued mapping and its applications in optimization. Ann Inst H Poincaré 6:449–482

    Article  MathSciNet  Google Scholar 

  • Shi DS (1991) Contingent derivative of the perturbation map in multiobjective optimization. J Optim Theory Appl 70:385–396

    Article  MathSciNet  Google Scholar 

  • Song W (1996) Duality for vector optimization of set-valued functions. J Math Anal Appl 201:212–225

    Article  MathSciNet  Google Scholar 

  • Sun XK, Li SJ (2011) Lower Studniarski derivative of the perturbation map in parametrized vector optimization. Optim Lett 5:601–614

    Article  MathSciNet  Google Scholar 

  • Taa A (1998) Set-valued derivatives of multifunctions and optimality conditions. Numer Funct Anal Optim 19:121–140

    Article  MathSciNet  Google Scholar 

  • Tanino T (1988a) Sensitivity analysis in multiobjective optimization. J Optim Theory Appl 56:479–499

    Article  MathSciNet  Google Scholar 

  • Tanino T (1988b) Stability and sensitivity analysis in convex vector optimization. SIAM J Control Optim 26:524–536

    Article  MathSciNet  Google Scholar 

  • Wang ED (2008) A chain rule for first and second order epiderivatives and hypoderivatives. J Math Anal Appl 348:324–336

    Article  MathSciNet  Google Scholar 

  • Wang QL, Li S (2012) Sensitivity and stability for the second-order contingent derivative of the proper perturbation map in vector optimization. Optim Lett 6:731–748

    Article  MathSciNet  Google Scholar 

  • Xu Y, Peng Z (2017) Higher-order sensitivity analysis in set-valued optimization under Henig efficiency. J Ind Manag Optim 13:313–327

    MathSciNet  MATH  Google Scholar 

  • Zhu SK, Li SJ, Teo KL (2014) Second-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization. J Glob Optim 58:673–692

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research was funded by Vietnam National University Ho Chi Minh City (VNU-HCMC) under grant number B2018-28-02. We are thankful to the anonymous referees for their useful comments to improve the manuscript.

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Authors and Affiliations

  1. Department of Optimization and System Theory, VNU HCM-University of Science, Ho Chi Minh City, Vietnam

    Nguyen Le Hoang Anh

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  1. Nguyen Le Hoang Anh
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Correspondence to Nguyen Le Hoang Anh.

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Communicated by Hector Ramirez.

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Anh, N.L.H. Second-order composed contingent derivatives of perturbation maps in set-valued optimization. Comp. Appl. Math. 38, 145 (2019). https://doi.org/10.1007/s40314-019-0923-4

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  • DOI: https://doi.org/10.1007/s40314-019-0923-4

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