Taiwanese Journal of Mathematics

$\mathbb{T}$-Epiderivatives of Set-valued Maps and Its Application to Set Optimization and Generalized Variational Inequalities

Qamrul Hasan Ansari and Johannes Jahn

Full-text: Open access

Abstract

In this paper, we first define a $\mathbb{T}$-cone which is a unified version of several cones, namely, contingent cone, radial cone, $C$-tangent cone, Clarke tangent cone, $S$-cone, adjacent cone, etc. Then, we define the $\mathbb{T}$-epiderivative of a set-valued map which includes the contingent epiderivative, radial epiderivative, $S$-epiderivative, adjacent epiderivative etc. as special cases. We present several properties of such an epiderivative. The generalized vector $\mathbb{T}$-variational inequality problem is also considered. We provide necessary and sufficient conditions for a solution of a set optimization problem. Several existence results for solutions of set optimization problems and a generalized vector $\mathbb{T}$-variational inequality problem are given.

Article information

Source
Taiwanese J. Math., Volume 14, Number 6 (2010), 2447-2468.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406084

Digital Object Identifier
doi:10.11650/twjm/1500406084

Mathematical Reviews number (MathSciNet)
MR2761608

Zentralblatt MATH identifier
1241.90131

Subjects
Primary: 90C29: Multi-objective and goal programming 49J40: Variational methods including variational inequalities [See also 47J20] 90C26: Nonconvex programming, global optimization 90C48: Programming in abstract spaces

Keywords
$\mathbb{T}$-cone $\mathbb{T}$-epiderivative contingent epiderivative radial epiderivative $S$-epiderivative optimality conditions vector optimization problem generalized vector $\mathbb{T}$-variational inequality problem existence results

Citation

Ansari, Qamrul Hasan; Jahn, Johannes. $\mathbb{T}$-Epiderivatives of Set-valued Maps and Its Application to Set Optimization and Generalized Variational Inequalities. Taiwanese J. Math. 14 (2010), no. 6, 2447--2468. doi:10.11650/twjm/1500406084. https://projecteuclid.org/euclid.twjm/1500406084


Export citation

References

  • J. P. Aubin, Contingent derivatives of set-valued maps and existence of soltuions to nonlinear inclusions and differential inclusions, in: Mathematical Analysis and Applications, Part A, L. Nachbin (ed.), Academic Press, New York, 1981, pp. 160-229.
  • J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, New York, 1984.
  • Q. H. Ansari and J. C. Yao, On nondifferentiable and nonconvex vector optimization problems, J. Optim. Theory Appl., 106 (2000), 487-500.
  • J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, Basel, Berlin, 1990.
  • G. Bigi and M. Castellani, $K$-epiderivatives for set-valued functions and optimization, Math. Meth. Oper. Res., 55 (2002), 401-412.
  • G. Bouligand, Sur l'existence des demi-tangentes a une courbe de Jordan, Fund. Math., 15 (1930), 215-218.
  • G. Y. Chen and B. D. Craven, A vector variational inequality and optimization over an efficient set, ZOR-Meth. Model. Oper. Res., 3 (1990), 1-12.
  • L. Chen, Generalized tangent epiderivative and applications to set-valued map optimization, J. Nonlinear Convex Anal., 3(3) (2002), 303-313.
  • H. W. Corley, Optimality conditions for maximizations of set-valued functions, J. Optim. Theory Appl., 58 (1988), 1-10.
  • F. Flores-Bazán, Optimality conditions in non-convex set-valued optimization, Math. Meth. Oper. Res., 53 (2001), 403-417.
  • F. Flores-Bazán, Radial epiderivatives and asymptotic functions in nonconvex vector optimization, SIAM J. Optim., 14(1) (2003), 284-305.
  • F. Giannessi, On Minty variational principle. in: New Trends in Mathematical Programming, F. Giannessi, S. Komloski and T. Tapcsáck (eds.), Kluwer Academic Publisher, Dordrech, The Netherlands, 1998, pp. 93-99.
  • J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Springer-Verlag, Berlin, Heidelberg, New York, 1993.
  • J. Jahn, Introduction to the Theory of Nonlinear Optimization, Springer-Verlag, Berlin, Heidelberg, 2007.
  • J. Jahn and A. A. Khan, Some calculus rules for contingent epiderivatives, Optimization, 52 (2003), 113-125.
  • J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimization, Math. Meth. Oper. Res., 46 (1997), 193-211.
  • S. Komlósi, On the Stampacchia and Minty variational inequalities, in: Generalized Convexity and Optimization for Economic and Financial Decisions, G. Giorgi and F. Rossi (eds.), Pitagora Editrice, Bologna, Italy, 1999, pp. 231-260.
  • C. S. Lalitha, J. Dutta and M. G. Govil, Optimality conditions in set-valued optimization, J. Austral. Math. Soc., 75 (2003), 1-11.
  • G. M. Lee, D. S. Kim and H. Kuk, Existence of solutions for vector optimization problems. J. Math. Anal. Appl., 220 (1998), 90-98.
  • G. M. Lee, D. S. Kim, B. S. Lee and N. D. Yen, Vector variational inequality as a tool for studying vector optimization problems, Nonlinear Anal., 34 (1998), 745-765.
  • D. T. Luc, Theory of Vector Optimization, Springer-Verlag, Berlin, 1989.
  • D. T. Luc, Contingent derivative of set-valued maps and applications to vector optimization, Math. Prog., 50 (1991), 99-111.
  • L.-J. Lin, Z.-T. Yu, Q. H. Ansari and L.-P. Lai, Fixed point and maximal element theorems with applications to abstract economies and minimax inequalities, J. Math. Anal. Appl., 284 (2003), 656-671.
  • W. V. Petryshyn and P. M. Fitzpatrick, Fixed point theorems of multivalued noncompact acyclic mappings, Pacific J. Math., 54 (1974), 17-23.
  • F. Severi, Su alcune questioni di topologia infinitesimale, Ann. Soc. Polon. Math., 9 (1930), 97-108.
  • D. S. Shi, Contingent derivative of the perturbation map in multiobjective optimization, J. Optim. Theory Appl., 70 (1991), 385-395.
  • A. Taa, Necessary and sufficient conditions for multiobjective optimization problems, Optimization, 36 (1996), 97-104.
  • A. Taa, Set-valued derivatives of multifunctions and optimality conditions, Numer. Funct. Anal. Optim., 19(1&2) (1998), 121-140.
  • D. Ward, Convex subcones of the contingent cone in nonsmooth calculus and optimization, Trans. Amer. Math. Soc., 302 (1987), 661-682.
  • D. E. Ward and G. M. Lee, On relations between vector optimization problems and vector variational inequalities, J. Optim. Theory Appl., 113(3) (2002), 583-596.