On second-order derivatives of the efficient point multifunction in parametric vector optimization problems

• Autorzy: Pham Thanh-Hung , Nguyen Thanh-Sang

Identyfikatory

DOI

10.2478/candc-2023-0043

• Języki publikacji: EN

Abstrakty

EN

In this paper, we establish formulae for inner and outer evaluation of the second-order contingent derivative of index nu of the efficient point multifunction in parametric vector optimization problems. The results contained in this paper extend the results of Chuong (2013a) to the second-order sensitivity analysis case. On the other hand, examples are provided for purposes of analyzing and illustrating the obtained results. Concerning the potential domain of application, the functioning of the majority of economic systems depends on a set of indicators (criteria), i.e., the substance of economic systems includes multiple criteria and only the lack of mathematical methods in solving the problems of vector optimization is an obstacle to the effective use of the respective models. Therefore, the study of vector optimization problems is necessary and has practical significance.

Słowa kluczowe

EN

parametric vector optimization; efficient point multifunction; second-order contingent derivative of index nu; sensitivity analysis

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2023
• Tom: Vol. 52, No. 4
• Strony: 381--403
• Opis fizyczny: Bibliogr. 50 poz.

Twórcy

autor

Pham Thanh-Hung

• Faculty of Pedagogy and Faculty of Social Sciences & Humanities, Kien Giang University, Kien Giang Province, Vietnam

autor

Nguyen Thanh-Sang

• Faculty of Pedagogy and Faculty of Social Sciences & Humanities, Kien Giang University, Kien Giang Province, Vietnam

Bibliografia

• Anh, N. L. H. and Khanh, P. Q. (2013) Variational sets of perturbation maps and applications to sensitivity analysis for constrained vector optimization. Journal of Optimization Theory and Applications, 158, 363-384.
• Anh, N. L. H. (2017a) Sensitivity analysis in constrained set-valued optimization via Studniarski derivatives. Positivity, 21, 255–272.
• Anh, N. L. H. (2017b) Some results on sensitivity analysis in set-valued optimization. Positivity, 21, 1527–1543.
• Anh, N. L. H. and Thoa, N. T. (2020) Calculus rules of the generalized contingent derivative and applications to set-valued optimization. Positivity, 24, pp. 81–94.
• Aubin, J. P. and Frankowska, H. (1990) Set-Valued Analysis. Birkhäuser, Boston.
• Bonnans, J. F. and Shapiro, A. (2000) Perturbation Analysis of Optimization Problems. Springer, New York.
• Chuong, T. D. (2011) Clarke coderivatives of efficient point multifunctions in parametric vector optimization. Nonlinear Analysis, 74, 273–285.
• Chuong, T. D. (2013a) Derivatives of the efficient point multifunction in parametric vector optimization problems. Journal of Optimization Theory and Applications. 156, 247-265.
• Chuong, T. D. (2013b) Normal subdifferentials of effcient point multifunctions in parametric vector optimization. Optimization Letters, 7, 1087–1117.
• Chuong, T. D. and Yao, J.-C. (2009) Coderivatives of efficient point multifunctions in parametric vector optimization. Taiwanese Journal of Mathematics, 13, 1671-1693.
• Chuong, T. D. and Yao, J.-C. (2010) Generalized Clarke epiderivatives of parametric vector optimization problems. Journal of Optimization Theory and Applications, 147, 77-94.
• Chuong, T. D. and Yao, J.-C. (2013a) Fréchet subdifferentials of efficient point multifunctions in parametric vector optimization. Journal of Global Optimization, 57, 1229–1243.
• Chuong, T. D. and Yao, J.-C. (2013b) Isolated calmness of efficient point multifunctions in parametric vector optimization. Journal of Nonlinear and Convex Analysis, 14, 719-734.
• Diem, H. T. H., Khanh, P. Q. and Tung, L. T. (2014) On higher-order sensitivity analysis in nonsmooth vector optimization. Journal of Optimization Theory and Applications, 162, 463–488.
• Huy, N. Q. and Lee, G. M. (2007) On sensitivity analysis in vector optimization. Taiwanese Journal of Mathematics, 11, 945-958.
• Huy, N. Q. and Lee, G. M. (2008) Sensitivity of solutions to a parametric generalized equation. Set-Valued and Variational Analysis, 16, 805–820.
• Khan, A., Tammer, C. and Zӑlinescu, C. (2015) Set-Valued Optimization: An Introduction with Applications. Springer, Berlin.
• Kuk, H., Tanino, T. and Tanaka, M. (1996) Sensitivity analysis in parameterized convex vector optimization. Journal of Mathematical Analysis and Applications, 202, 511–522.
• Lee, G. M. and Huy, N. Q. (2006) On proto-differentiability of generalized perturbation maps. Journal of Mathematical Analysis and Applications, 324, 1297–1309.
• Levy, A. B. and Rockafellar, R. T. (1994) Sensitivity analysis of solutions to generalized equations. Transactions of the American Mathematical Society, 345, 661–671.
• Li, S. J. and Liao, C. M. (2012) Second-order differentiability of generalized perturbation maps. Journal of Global Optimization, 52, 243–252.
• Li, S. J., Sun, X. K. and Zhai, J. (2012) Second-order contingent derivatives of set-valued mappings with application to set-valued optimization. Applied Mathematics and Computation, 218, 6874–6886.
• Luc, D. T. (1989) Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems. 319, Springer-Verlag, Berlin.
• Luc, D. T., Soleimani-damaneh, M. and Zamani, M. (2018) Semi-differentiability of the marginal mapping in vector optimization. SIAM Journal on Optimization, 28, 1255–1281.
• Mordukhovich, B. S. (2006a) Variational Analysis and Generalized Differentiation. I: Basic Theory. Springer, Berlin.
• Mordukhovich, B. S. (2006b) Variational Analysis and Generalized Differentiation. II: Applications. Springer, Berlin.
• Mordukhovich, B. S. (2018) Variational Analysis and Applications. Springer, New York.
• Pham, T. H. (2022) On generalized second-order proto-differentiability of the Benson proper perturbation maps in parametric vector optimization problems. Positivity, 26, 1-36.
• Pham, T. H. (2023a) On second-order semi-differentiability of index of perturbation maps in parametric vector optimization problems. Asia-Pacific Journal of Operational Research, 40, 1-38.
• Pham, T. H. (2023b) Generalized higher-order semi-derivative of the perturbation maps in vector optimization, Japan Journal of Industrial and Applied Mathematics , 40, 929–963.
• Pham, T.H and Nguyen, T. S. (2022) On second-order radial-asymptotic proto-differentiability of the Borwein perturbation maps. RAIRO - Operations Research, 56, 1373–1395.
• Rockafellar, R. T. (1989) Proto-differentiablility of set-valued mappings and its applications in optimization. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 6, 449–482.
• Rockafellar, R. T. and Wets, R. J. B. (2009) Variational Analysis. Springer, Berlin.
• Shi, D. S. (1991), Contingent derivative of the perturbation map in multiobjective optimization. Journal of Optimization Theory and Applications, 70, 385–396.
• Shi, D. S. (1993) Sensitivity analysis in convex vector optimization. Journal of Optimization Theory and Applications, 77, 145–159.
• Sun, X. K. and Li, S. J. (2011) Lower Studniarski derivative of the perturbation map in parametrized vector optimization. Optimization Letters, 5, 601–614.
• Sun, X. K. and Li, S. J. (2014) Generalized second-order contingent epiderivatives in parametric vector optimization problem. Journal of Global Optimization, 58, 351-363.
• Tanino, T. (1988a) Sensitivity analysis in multiobjective optimization, Journal of Optimization Theory and Applications. 56, 479–499.
• Tanino, T. (1988b) Stability and sensitivity analysis in convex vector optimization. SIAM Journal on Control and Optimization, 26, 521–536.
• Tung, L. T. (2017a) Second-order radial-asymptotic derivatives and applications in set-valued vector optimization. Pacific Journal of Optimization, 13, 137–153.
• Tung, L. T. (2017b) Variational sets and asymptotic variational sets of proper perturbation map in parametric vector optimization. Positivity, 21, 1647–1673.
• Tung, L. T. (2018) On second-order proto-differentiability of perturbation maps. Set-Valued and Variational Analysis, 26, 561–579.
• Tung, L. T. (2020) On higher-order proto-differentiability of perturbation maps. Positivity, 24, 441–462.
• Tung, L. T. (2021) On higher-order proto-differentiability and higher-order asymptotic proto-differentiability of weak perturbation maps in parametric vector optimization. Positivity, 25, 579–604.
• Tung, L. T. (2021) On second-order composed proto-differentiability of proper perturbation maps in parametric vector optimization problems. Asia-Pacific Journal of Operational Research, 38, 2050040.
• Tung, L. T. and Pham, T. H. (2020a) Sensitivity analysis in parametric vector optimization in Banach spaces via _w−contingent derivatives. Turkish Journal of Mathematics, 44, 152–168.
• Tung, L. T. and Pham, T. H. (2020b) On generalized _w-contingent epiderivatives in parametric vector optimization problems. Applied Set-Valued Analysis and Optimization, 2, 152–168.
• Wang, Q. L., Li, S. J and Teo, K. L. (2010) Higher-order optimality conditions for weakly efficient solutions in nonconvex set-valued optimization. Optimization Letters, 4, 425–437.
• Wang, Q. L. and Li, S. J. (2011) Second-order contingent derivative of the perturbation map in multiobjective optimization. Journal of Fixed Point Theory and Applications, 857520.
• Wang, Q. L. and Li, S. J. (2012) Sensitivity and stability for the secondorder contingent derivative of the proper perturbation map in vector optimization. Optimization Letters, 6, 731–748.