Taiwanese Journal of Mathematics


N. Q. Huy, B. S. Mordukhovich, and J. C. Yao

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This paper concerns sensitivity analysis for general parametric constrained problems of multiobjective optimization in infinite-dimensional spaces by using advanced tools of modern variational analysis and generalized differentiation. We pay the main attention to computing and estimating coderivatives of frontier and efficient solution maps in parametric multiobjective problems with respect to generalized order optimality that include a vast majority of conventional multiobjective problems in the presence of geometric, operator, functional, and equilibrium constraints. The obtained results are new in both finite-dimensional and infinite-dimensional spaces.

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Taiwanese J. Math., Volume 12, Number 8 (2008), 2083-2111.

First available in Project Euclid: 18 July 2017

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Zentralblatt MATH identifier

Primary: 90C29: Multi-objective and goal programming 90C30: Nonlinear programming 49J52: Nonsmooth analysis [See also 46G05, 58C50, 90C56] 49J53: Set-valued and variational analysis [See also 28B20, 47H04, 54C60, 58C06]

variational analysis parametric multiobjective optimization generalized order and generalized Pareto optimality frontier and efficient solution maps set-valued mappings coderivatives Lipschitzian properties


Huy, N. Q.; Mordukhovich, B. S.; Yao, J. C. CODERIVATIVES OF FRONTIER AND SOLUTION MAPS IN PARAMETRIC MULTIOBJECTIVE OPTIMIZATION. Taiwanese J. Math. 12 (2008), no. 8, 2083--2111. doi:10.11650/twjm/1500405137. https://projecteuclid.org/euclid.twjm/1500405137

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  • T. Q. Bao and B. S. Mordukhovich, Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Prog., Ser. A, (2008), to appear.
  • E. M. Bednarczuk, Weak sharp efficiency and growth condition for vector-valued functions with applications. Optimization, 53 (2004), 455-474.
  • J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 2000.
  • J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis. Springer, New York, (2005).
  • M. Durea, Optimality conditions for weak and firm efficiency in set-valued optimization. J. Math. Anal. Appl., (2008), to appear.
  • J. Dutta and C. Tammer, Lagrangian conditions for vector optimization in Banach spaces. Math. Meth. Oper. Res., 64 (2006), 521-541.
  • M. I. Henig, The domination property in multicriteria optimization. J. Math. Anal. Appl., 114 (1986), 7-16.
  • R. Henrion, Characterization of stability for cone increasing constraints in stochastic programming. Set-Valued Anal., 5 (1997), 323-349.
  • J. Jahn, Vector Optimization: Theory, Applications and Extensions, Springer, New York, 2004.
  • V. Jeyakumar and D. T. Luc, Nonsmooth Vector Functions and Continuous Optimization, Springer, New York, 2008.
  • G. M. Lee and N. Q. Huy, On sensitivity analysis in vector optimization. Taiwanese J. Math., 11 (2007), 945-958.
  • A. B. Levy and B. S. Mordukhovich, Coderivatives in parametric optimization, Math. Prog., Ser. A, 99 (2004), 311-327.
  • D. T. Luc, Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, 319, Springer-Verlag, Berlin, 1989.
  • D. T. Luc, Contingent derivatives of set-valued maps and applications to vector optimization. Math. Prog., Ser. A, 50 (1991), 99-111.
  • Y. Lucet and J. J. Ye, Sensitivity analysis of the value function for optimization problems with variational inequality constraints, SIAM J. Control Optim., 40 (2001), 699-723; Erratum SIAM J. Control Optim., 41 (2002), 1315-1319.
  • B. S. Mordukhovich, Maximum principle in problems of time optimal control with nonsmooth constraints, J. Appl. Math. Mech., 40 (1976), 960-969.
  • B. S. Mordukhovich, Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problmes, Soviet Math. Dokl., 22 (1980), 526-530.
  • B. S. Mordukhovich, Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions, Trans. Amer. Math. Soc. 340 (1993), 1-35.
  • B. S. Mordukhovich, Coderivative of set-valued mappings: calculus and applications, Nonlinear Anal., 30 (2004), 3059-3070.
  • B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, Springer, Berlin, 2006.
  • B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, II: Applications, Springer, Berlin, 2006.
  • B. S. Mordukhovich, Multiobjective optimization problems with equilibrium constraints, Math. Prog., Ser. B, (2008), (appeared in the online first list).
  • B. S. Mordukhovich, N. M. Nam and N. D. Yen, Subgradients of marginal functions in parametric mathematical programming. Math. Prog., Ser. B, 116 (2009), 369-396.
  • B. S. Mordukhovich and J. V. Outrata, Coderivative analysis of quasi-variational inequalities with applications to stability and optimization, SIAM J. Optim., 18 (2007), 389-412.
  • J. V. Outrata, Optimality conditions for a class of mathematical programs with equilibrium constraints. Math. Oper. Res., 24 (1999), 627-644.
  • J. V. Outrata, A note on a class of equilibrium problems with equilibrium constraints, Kybernetika, 40 (2004), 585-594.
  • S. M. Robinson, Generalized equations and their solutions, I: Basic theory, Math. Progr. Study, 10 (1979), 128-141.
  • R. T. Rockafellar and R. J-B. Wets, Variational Analysis, Springer, Berlin, 1998.
  • D. S. Shi, Contingent derivative of the perturbation map in multiobjective optimization, J. Optim. Theory Appl., 70 (1991), 385-396.
  • W. Schirotzek, Nonsmooth Analysis, Springer, Berlin, 2007.
  • T. Tanino, Sensitivity analysis in multiobjective optimization, J. Optim. Theory Appl., 56 (1988), 479-499.
  • J. J. Ye and Q. J. Zhu, Multiobjective optimization problem with variational inequality constraints, Math. Prog., Ser. A , 96 (2003), 139-160. \def\ucc$\breve{\rm a}$
  • C. Tammer and C. Z\ucc linescu, Minimal elements for product spaces, Optimization, (2008), to appear.
  • R. Zhang, Weakly upper Lipschitzian multifunctions and applications to parametric optimization, Math. Prog., Ser. A, 102 (2005), 153-166.
  • X. Y. Zhang and K. F. Ng, The Fermat rule for multifunctions in Banach spaces, Math. Prog., Ser. A, 104 (2005), 69-90.