Abstract
Here we start studying problems of set optimization and interrelated ones of multiobjective optimization, where optimal solutions are understood in various Pareto-type sense with respect to general preference relations.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
S. Adly and A. Seeger (2011), A nonsmooth algorithm for cone-constrained eigenvalue problems, Comput. Optim. Appl. 49, 299–318.
S. Al-Homidan, Q. H. Ansari and J.-C. Yao (2008), Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory, Nonlinear Anal. 69, 126–139.
Q. H. Ansari, C. S. Lalitha and M. Mehta (2014), Generalized Convexity, Nonsmooth variational inequalities, and Nonsmooth Optimization, CRC Press, Boca Raton, Florida.
T. Q. Bao, P. Gupta and B. S. Mordukhovich (2007), Necessary conditions for multiobjective optimization with equilibrium constraints, J. Optim. Theory Appl. 135, 179–203.
T. Q. Bao and B. S. Mordukhovich (2007), Variational principles for set-valued mappings with applications to multiobjective optimization, Control Cybern. 36, 531–562.
T. Q. Bao and B. S. Mordukhovich (2007), Existence of minimizers and necessary conditions in set-valued optimization with equilibrium constraints, Appl. Math. 52, 453–472.
T. Q. Bao and B. S. Mordukhovich (2009), Necessary conditions for super minimizers in constrained multiobjective optimization, J. Global Optim. 43, 533–552.
T. Q. Bao and B. S. Mordukhovich (2010), Relative Pareto minimizers for multiobjective problems: existence and optimality conditions, Math. Program 122, 301–347.
T. Q. Bao and B. S. Mordukhovich (2010), Set-valued optimization in welfare economics, Adv. Math. Econ. 13, 114–153.
T. Q. Bao and B. S. Mordukhovich (2011), Refined necessary conditions in multiobjective optimization with applications to microeconomic modeling, Discrete Contin. Dyn. Syst. 31, 1069–1096.
T. Q. Bao and B. S. Mordukhovich (2012), Sufficient conditions for global weak Pareto solutions in multiobjective optimization, Positivity 16, 579–602.
T. Q. Bao and B. S. Mordukhovich (2012), Extended Pareto optimality in multiobjective problems, in Recent Developments in Vector Optimization, edited by Q. H. Ansari and J.-C. Yao, pp. 467–515, Springer, Berlin.
T. Q. Bao and B. S. Mordukhovich (2014), Necessary nondomination conditions in set and vector optimization with variable ordering structures, J. Optim. Theory Appl. 162, 350–370.
T. Q. Bao, B. S. Mordukhovich and A. Soubeyran (2015), Variational analysis in psychological modeling, J. Optim. Theory Appl. 164, 290–315.
T. Q. Bao, B. S. Mordukhovich and A. Soubeyran (2015), Fixed points and variational principles with applications to capability theory of wellbeing via variational rationality, Set-Valued Var. Anal. 23, 375–398.
T. Q. Bao, B. S. Mordukhovich and A. Soubeyran (2015), Minimal points, variational principles, and variable preferences in set optimization, J. Nonlinear Convex Anal. 16, 1511–1537.
T. Q. Bao and C. Tammer (2012), Lagrange necessary conditions for Pareto minimizers in Asplund spaces and applications, Nonlinear Anal. 75, 1089–1103.
A. Barbagallo and P. Mauro (2016), A general quasi-variational problem of Cournot-Nash type and its inverse formulation, J. Optim. Theory Appl. 170 (2016), 476–492.
E. M. Bednarczuk and D. Zagrodny (2009), Vector variational principle, Arch. Math. 93, 577–586.
S. Bellaassali and A. Jourani (2008), Lagrange multipliers for multiobjective programs with a general preference, Set-Valued Anal. 16, 229–243.
M. Bianchi, N. Hadjisavvas and S. Schaible (1997), Vector equilibrium problems with generalized monotone bifunctions, J. Optim. Theory Appl. 92, 527–542.
G. Bigi, M. Castellani, M. Pappalardo and M. Passacantando (2013), Existence and solution methods for equilibria, European J. Oper. Res. 227, 1–11.
E. Blum and W. Oettli (1994), From optimization and variational inequalities to equilibrium problems, Math. Student 63, 123–145.
H. Bonnel, A. N. Iusem and B. F. Svaiter (2005), Proximal methods in vector optimization, SIAM J. Optim. 15, 953–970.
J. M. Borwein and R. Goebel (2003), Notions of relative interior in Banach spaces, J. Math. Sci. 115, 2542–2553.
J. M. Borwein and A. S. Lewis (1992), Partially finite convex programming, I: quasi relative interiors and duality theory, Math. Prog. 57, 15–48.
J. M. Borwein, Y. Lucet and B. S. Mordukhovich (2000), Compactly epi-Lipschitzian convex sets and functions in normed spaces, J. Convex Anal. 7, 375–393.
J. M. Borwein and D. Zhuang (1993), Super efficiency in vector optimization, Trans. Amer. Math. Soc. 338, 105–122.
R. I. Boţ (2010), Conjugate Duality in Convex Optimization, Springer, Berlin.
R. I. Boţ (2012), An upper estimate for the Clarke subdifferential of an infimal value function proved via the Mordukhovich subdifferential, Nonlinear Anal. 75, 1141–1146.
R. I. Boţ, E. R. Csetnek and G. Wanka (2008), Regularity conditions via quasi-relative interior in convex programming, SIAM J. Optim. 19, 217–233.
R. I. Boţ, S. M. Grad and G. Wanka (2008), New regularity conditions for strong and total Fenchel-Lagrange duality in infinite-dimensional spaces, Nonliner Anal. 69, 323–336.
L. C. Ceng, B. S. Mordukhovich and J.-C. Yao (2010), Hybrid approximate proximal method with auxiliary variational inequality for vector optimization, J. Optim. Theory Appl. 146 (2010), 267–303.
C. R. Chen, S. J. Li and K. L. Teo (2009), Solution semicontinuity of parametric generalized vector equilibrium problems, J. Global Optim. 45, 309–318.
G. Y. Chen, X. X. Huang and X. Q. Yang (2005), Vector Optimization, Springer, Berlin.
T. D. Chuong, B. S. Mordukhovich and J.-C. Yao (2011), Hybrid approximate proximal algorithms for efficient solutions in vector optimization, J. Nonlinear Convex Anal. 12, 257–286.
S. Cobzaş (2013), Functional Analysis in Asymmetric Normed Spaces, Birkhüser, Basel, Switzerland.
H. W. Corley (1988), Optimality conditions for maximization of set-valued functions, J. Optim. Theory Appl. 58, 1–10.
G. P. Crespi, I. Ginchev and M. Rocca (2006), First-order optimality conditions in set-valued optimization, Math. Methods Oper. Res. 63, 87–106.
P. Daniele, S. Giuffré, G. Idone and A. Maugeri (2007), Infinite dimensional duality and applications, Math. Ann. 339, 221–239.
M. B. Donato, M. Milasi and C. Vitanza (2008), An existence result of a quasi-variational inequality associated to an equilibrium problem, J. Global Optim. 40, 87–97.
M. Durea, J. Dutta and C. Tammer (2010), Lagrange multipliers and ɛ-Pareto solutions in vector optimization with nonsolid cones in Banach spaces, J. Optim. Theory Appl. 145, 196–211.
M. Durea and R. Strugariu (2010), Necessary optimality conditions for weak sharp minima in set-valued optimization, Nonlinear Anal. 73, 2148–2157.
J. Dutta (2005), Optimality conditions for maximizing a locally Lipschitz function, Optimization 54, 377–389.
J. Dutta (2012), Strong KKT, second order conditions and non-solid cones in vector optimization, in Recent Developments in Vector Optimization, edited by Q. H. Ansari and J.-C. Yao, pp. 127–167, Springer, Berlin.
M. Ehrgott (2005), Multicriteria Optimization, 2nd edition, Springer, Berlin.
G. Eichfelder (2014), Variable Ordering Structures in Vector Optimization, Springer, Berlin.
G. Eichfelder and T. X. D. Ha (2013), Optimality conditions for vector optimization problems with variable ordering structures, Optimization 62, 1468–1476.
G. Eichfelder and R. Kasimbeyli (2013), Properly optimal elements in vector optimization with variable ordering structures, J. Global Optim. 60, 689–712.
B. El Abdoini and L. Thibault (1996), Optimality conditions for problems with set-valued objectives, J. Appl. Anal. 2, 183–201.
E.-A. Florea (2016), Coderivative necessary optimality conditions for sharp and robust efficiencies in vector optimization with variable ordering structure, Optimization 65 (2016), 1417–1435.
F. Flores-Bazán and E. Hernández (2013), Optimality conditions for a unified vector optimization problem with not necessarily preordering relations, J. Global Optim. 56, 299–315.
M. Fukushima and J.-S. Pang (2005), Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games, Comput. Management Sci. 1, 21–56.
C. Gerth (Tammer) and P. Weidner (1990), Nonconvex separation theorems and some applications in vector optimization, J. Optim. Theory Appl. 67, 297–320.
F. Giannessi (1980), Theorems of alternative, quadratic programs and complementarity problems, in Variational Inequalities and Complementarity Problems, edited by R. Cottle et al., pp. 151–186, Wiley, New York.
F. Giannessi (2005), Constrained Optimization and Image Space Analysis, Springer, London, United Kingdom.
A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu (2003), Variational Methods in Partially Ordered Spaces, Springer, New York.
V. V. Gorokhovik (1990), Convex and Nonsmooth Problems of Vector Optimization, Nauka i Tekhnika, Minsk, Belarus.
S.-M. Grad (2015), Vector Optimization and Monotone Operators via Convex Duality, Springer, Berlin.
L. M. Graña Drummond and B. F. Svaiter (2005), A steepest descent method for vector optimization, J. Comput. Appl. Math. 175 (2005), 395–414.
A. Guerraggio and D. T. Luc (2006), Properly maximal points in product spaces, Math. Oper. Res. 31, 305–315.
C. Gutiérrez, L. Huerga, B. Jiménez and V. Novo (2013), Proper approximate solutions and ɛ-subdifferentials in vector optimization: basic properties and limit behaviour, Nonlinear Anal. 79, 52–67.
T. X. D. Ha (2005), Lagrange multipliers for set-valued problems associated with coderivatives, J. Math. Anal. Appl. 311, 647–663.
T. X. D. Ha (2005), Some variants of the Ekeland variational principle for a set-valued map, J. Optim. Theory Appl. 124, 187–206.
T. X. D. Ha (2012), Optimality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems, Nonlinear Anal. 75, 1305–1323.
T. X. D. Ha (2012), The Fermat rule and Lagrange multiplier rule for various effective solutions to set-valued optimization problems expressed in terms of coderivatives, in Recent Developments in Vector Optimization, edited by Q. H. Ansari and J.-C. Yao, pp. 417–466, Springer, Berlin.
N. Hadjisavvas (2005), Generalized convexity, generalized monotonicity and nonsmooth analysis, Handbook of Generalized Convexity and Generalized Monotonicity, edited by N. Hadjisavvas et al., pp. 465–499, Springer, New York.
A. H. Hamel, F. Heyde, A. Löhne, B. Rudloff and C. Schrage (2015), Set optimization–a rather short introduction, in Set Optimization and Applications–the State of the Art, edited by A. H. Hamel et al., pp. 65–141, Springer, Berlin.
A. H. Hamel and C. Schrage (2014), Directional derivatives, subdifferentials and optimality conditions for set-valued convex functions, Pac. J. Optim. 10, 667–689.
E. Hernández, L. Rodríguez-Marín and M. Sama (2010), On solutions of set-valued optimization problems, Comput. Math. Appl. 65, 1401–1408.
J.-B. Hiriart-Urruty and Y. S. Ledyaev (1996), A note on the characterization of the global maxima of a (tangentially) convex function over a convex set, J. Convex Anal. 3, 55–61.
X. Hu and D. Ralph (2007), Using EPECs to model bilevel games in restructured electricity markets with locational prices, Oper. Res. 55, 809–827.
H. Huang (2008), The Lagrange multiplier rule for super efficiency in vector optimization, J. Math. Anal. Appl. 342, 503–513.
N. Q. Huy, B. S. Mordukhovich and J.-C. Yao (2008), Coderivatives of frontier and solution maps in parametric multiobjective optimization, Taiwanese J. Math. 12, 2083–2111.
J. Ide, E. Köbis, D. Kuroiwa, A. Schöbel and C. Tammer (2014), The relationships between multicriteria robustness concepts and set-valued optimization, Fixed Point Theory Appl. 83.
A. N. Iusem and V. Sosa (2010), On the proximal point method for equilibrium problems in Hilbert spaces, Optimization 59, 1259–1274.
J. Jahn (2004), Vector Optimization: Theory, Applications and Extensions, Springer, Berlin.
J. Jahn and A. A. Khan (2003), Some calculus rules for contingent epiderivatives, Optimization 52, 113–125.
V. Jeyakumar and D. T. Luc (2008), Nonsmooth Vector Functions and Continuous Optimization, Springer, New York.
A. A. Khan, C. Tammer and C. Zălinescu (2015), Set-Valued Optimization. An Introduction with Applications, Springer, Berlin.
P. Q. Khanh and D. N. Quy (2011), On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings, J. Global Optim. 49 (2011), 381–396.
I. V. Konnov (2001), Combined Relaxation Methods for Variational Inequalities, Springer, Berlin.
A. Y. Kruger (1985), Generalized differentials of nonsmooth functions and necessary conditions for an extremum, Siberian Math. J. 26, 370–379.
A. Y. Kruger and B. S. Mordukhovich (1980), Extremal points and the Euler equation in nonsmooth optimization, Dokl. Akad. Nauk BSSR 24, 684–687 (in Russian).
D. Kuroiwa (1998), The natural criteria in set-valued optimization, RIMS Kokyuroku 1031, 85–90.
A. G. Kusraev and S. S. Kutateladze (1995), Subdifferentials: Theory and Applications, Kluwer, Dordrecht, The Netherlands.
G. Li, K. F. Ng and X. Y. Zheng (2007), Unified approach to some geometric results in variational analysis, J. Funct. Anal. 248 (2007), 317–343.
A. Löhne (2011), Vector Optimization with Infimum and Supremum, Springer, Berlin.
D. T. Luc (1989), Theory of Vector Optimization, Springer, Berlin.
G. Mastroeni (2003), Gap functions for equilibrium problems, J. Global Optim. 27, 411–426.
A. Maugeri (2001), Equilibrium problems and variational inequalities, in Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Methods, edited by F. Giannessi et al., pp. 187–205, Kluwer, Dordrecht, The Netherlands.
A. Maugeri and D. Puglisi (2014), A new necessary and sufficient condition for the strong duality and the infinite dimensional Lagrange multiplier rule, J. Math. Anal. Appl. 415, 661–676.
E. Miglierina, E. Molho and M. Rocca (2005), Well-posedness and scalarization in vector optimization, J. Optim. Theory Appl. 126, 391–409.
S. K. Mishra and V. Laha (2016), On Minty variational principle for nonsmooth vector optimization problems with approximate convexity, Optim. Lett. 10, 577–589.
B. S. Mordukhovich (1985), On necessary conditions for an extremum in nonsmooth optimization, Soviet Math. Dokl. 32, 215–220.
B. S. Mordukhovich (1988), Approximation Methods in Problems of Optimization and Control, Nauka, Moscow (in Russian).
B. S. Mordukhovich (2004), Equilibrium problems with equilibrium constraints via multiobjective optimization, Optim. Meth. Soft. 19, 479–492.
B. S. Mordukhovich (2006), Variational Analysis and Generalized Differentiation, I: Basic Theory, Springer, Berlin.
B. S. Mordukhovich (2006), Variational Analysis and Generalized Differentiation, II: Applications, Springer, Berlin.
B. S. Mordukhovich (2009), Methods of variational analysis in multiobjective optimization, Optimization 58, 413–430.
B. S. Mordukhovich (2009), Multiobjective optimization problems with equilibrium constraints, Math. Program. 117, 331–354.
B. S. Mordukhovich, N. M. Nam and N. D. Yen (2009), Subgradients of marginal functions in parametric mathematical programming, Math. Program. 116, 369–396.
B. S. Mordukhovich, J. V. Outrata and M. Černinka (2007), Equilibrium problems with complementarity constraints: case study with applications to oligopolistic markets, Optimization 56, 479–494.
B. S. Mordukhovich, B. Panicucci, M. Passacantando and M. Pappalardo (2012), Hybrid proximal methods for equilibrium problems, Optim. Lett. 6, 1535–1550.
B. S. Mordukhovich, J. S. Treiman and Q. J. Zhu (2003), An extended extremal principle with applications to multiobjective optimization, SIAM J. Optim. 14, 359–379.
B. S. Mordukhovich and B. Wang (2002), Necessary optimality and suboptimality conditions in nondifferentiable programming via variational principles, SIAM J. Control Optim. 41, 623–640.
B. S. Mordukhovich and B. Wang (2008) Generalized differentiation of parameter-dependent sets and mappings, Optimization 57, 17–40.
N. Nadezhkina and W. Takahashi (2006), Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim. 16, 1230–1241.
A. Nagurney (1999), Network Economics: A Variational Inequality Approach, 2nd edition, Kluwer, Dordrecht, The Netherlands.
W. Oettli (1982), Optimality conditions for programming problems involving multivalued mappings, in Modern Applied Mathematics, edited by B. H. Korte, pp. 195–226, North-Holland, Amsterdam.
J. V. Outrata, F. C. Ferris, M. Červinka and M. Outrata (2016), On Cournot-Nash-Walras equilibria and their computation, Set-Valued Var. Anal 24, 387–402.
M. Oveisiha and J. Zafarani (2012), Vector optimization problem and generalized convexity, J. Global Optim. 52, 29–43.
D. Pallaschke and S. Rolewicz (1998), Foundations of Mathematical Optimization: Convex Analysis without Linearity, Kluwer, Dordrecht, The Netherlands.
N. Popovici (2007), Explicitly quasiconvex set-valued optimization, J. Global Optim. 38, 103–118.
J.-H. Qiu (2014), A pre-order principle and set-valued Ekeland variational principle, J. Math. Anal. Appl. 419, 904–937.
M. Soleimani-damaneh (2010), Nonsmooth optimization using Mordukhovich’s subdifferential, SIAM J. Control Optim. 48, 3403–3432.
A. Soubeyran (2009), Variational rationality, a theory of individual stability and change: worthwhile and ambidextry behaviors, mimeo, GREQAM, Aix-Marseille University, France.
A. Taa (2011), On subdifferential calculus for set-valued mappings and optimality conditions, Nonlinear Anal. 74, 7312–7324.
S. Tagawa (1978), Optimierung mit Mengenwertigen Abbildugen, Ph.D. dissertation, University of Mannheim, Germany (in German).
N. V. Tuyen (2016), Some characterizations of solution sets of vector optimization problems with generalized order, Acta Math. Vietnamica 41, 677–694.
N. V. Tuyen and N. D. Yen (2012), On the concept of generalized order optimality, Nonlinear Anal. 75, 1592–1601.
D. E. Ward and G. M. Lee (2002), On relations between vector optimization problems and vector variational inequalities, J. Optim. Theory Appl. 113 (2002), 583–596.
X. Xue and Y. Zhang (2015), Coderivatives of gap function for Minty vector variational inequality, J. Inequal. Appl. 2015:285.
P. L. Yu (1974), Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives, J. Optim. Theory Appl. 14, 319–377.
A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM J. Control Optim. 42, 1071–1086.
C. Zălinescu (2015), On the use of the quasi-relative interior in optimization, Optimization 64, 1795–1823.
X. Y. Zheng and K. F. Ng (2005), The Fermat rule for multifunctions in Banach spaces, Math. Program. 104, 69–90.
X. Y. Zheng and K. F. Ng (2006), The Lagrange multiplier rule for multifunctions in Banach spaces, SIAM J. Optim. 17, 1154–1175.
S. K. Zhu (2016), Weak sharp efficiency in multiobjective optimization, Optim. Lett. 10, 1287–1301.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Mordukhovich, B.S. (2018). Variational Analysis in Set Optimization. In: Variational Analysis and Applications. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-92775-6_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-92775-6_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-92773-2
Online ISBN: 978-3-319-92775-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)