A note on Minty type vector variational inequalities
RAIRO - Operations Research - Recherche Opérationnelle, Tome 39 (2005) no. 4, pp. 253-273.

The existence of solutions to a scalar Minty variational inequality of differential type is usually related to monotonicity property of the primitive function. On the other hand, solutions of the variational inequality are global minimizers for the primitive function. The present paper generalizes these results to vector variational inequalities putting the Increasing Along Rays (IAR) property into the center of the discussion. To achieve that infinite elements in the image space $Y$ are introduced. Under quasiconvexity assumptions we show that solutions of generalized variational inequality and of the primitive optimization problem are equivalent. Finally, we discuss the possibility to generalize the scheme of this paper to get further applications in vector optimization.

DOI : https://doi.org/10.1051/ro:2006005
Classification : 47J20,  49J52,  90C29
Mots clés : minty vector variational inequality, existence of solutions, increasing-along-rays property, vector optimization
@article{RO_2005__39_4_253_0,
author = {Crespi, Giovanni P. and Ginchev, Ivan and Rocca, Matteo},
title = {A note on Minty type vector variational inequalities},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {253--273},
publisher = {EDP-Sciences},
volume = {39},
number = {4},
year = {2005},
doi = {10.1051/ro:2006005},
mrnumber = {2208753},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ro:2006005/}
}
Crespi, Giovanni P.; Ginchev, Ivan; Rocca, Matteo. A note on Minty type vector variational inequalities. RAIRO - Operations Research - Recherche Opérationnelle, Tome 39 (2005) no. 4, pp. 253-273. doi : 10.1051/ro:2006005. http://www.numdam.org/articles/10.1051/ro:2006005/

[1] C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, Applications to Free-Boundary Problems. John Wiley & Sons, New York (1984). | MR 745619 | Zbl 0551.49007

[2] G.P. Crespi, I. Ginchev and M. Rocca, Minty vector variational inequality, efficiency and proper efficiency. Vietnam J. Math. 32 (2004) 95-107. | Zbl 1056.49009

[3] G.P. Crespi, I. Ginchev and M. Rocca, Variational inequalities in vector optimization, in Variational Analysis and Applications, Series: Nonconvex Optimization and its Applications 79, edited by F. Giannessi and A. Maugeri. Springer, New York (2005), Part II, 259-278. | Zbl 1132.90015

[4] G.P. Crespi, I. Ginchev and M. Rocca, Minty variational inequalities, increase along rays property and optimization. J. Optim. Theory Appl. 123 (2004) 479-496. | Zbl 1059.49010

[5] G.P. Crespi, I. Ginchev and M. Rocca, Increase-along-rays property for vector functions, Preprint 2004/24, Universitá dell'Insubria, Facoltá di Economia, Varese, 2004, (http://eco.uninsubria.it/dipeco/Quaderni/files/QF2004_24.pdf).

[6] F. Giannessi, Theorems of alternative, quadratic programs and complementarity problems, in Variational Inequalities and Complementarity Problems, edited by R.W. Cottle, F. Giannessi and J.-L. Lions, John Wiley & Sons, New York (1980) 151-186. | Zbl 0484.90081

[7] F. Giannessi, On Minty variational principle, in New Trends in Mathematical Programming, edited by F. Giannessi, S. Komlósi, and T. Rapcsák, Kluwer, Dordrecht (1998) 93-99. | Zbl 0909.90253

[8] I. Ginchev, Higher order optimality conditions in nonsmooth vector optimization, in Generalized Convexity, Generalized Monotonicity, Optimality Conditions and Duality in Scalar and Vector Optimization, edited by A. Cambini, B.K. Dass and L. Martein. J. Stat. Manag. Syst. 5 (2002) 321-339. | Zbl 1079.90596

[9] I. Ginchev, A. Guerraggio and M. Rocca, First-order conditions for ${C}^{0,1}$ constrained vector optimization, in Variational Analysis and Applications, Series: Nonconvex Optimization and its Applications 79, edited by F. Giannessi and A. Maugeri. Springer, New York (2005), Part II, 437-450.

[10] J.-B. Hiriart-Urruty, New concepts in nondifferentiable programming, Analyse non convexe. Bull. Soc. Math. France 60 (1979) 57-85. | Numdam | Zbl 0469.90071

[11] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York (1980). | MR 567696 | Zbl 0457.35001

[12] A.N. Kolmogorov and S.V. Fomin, Elements of the theory of functions and of functional analysis, Nauka, Moscow (1972) (In Russian). | Zbl 0235.46001

[13] S. Komlósi, On the Stampacchia and Minty variational inequalities, in Generalized Convexity and Optimization for Economic and Financial Decisions, Proc. Verona, Italy, May 28-29, 1998, edited by G. Giorgi and F. Rossi, Pitagora Editrice, Bologna (1999) 231-260. | Zbl 0989.47055

[14] D.T. Luc, Theory of Vector Optimization. Springer-Verlag, Berlin (1989). | MR 1116766

[15] G. Mastroeni, Some remarks on the role of generalized convexity in the theory of variational inequalities, in Generalized Convexity and Optimization for Economic and Financial Decisions. Proc. Verona, Italy, May 28-29, 1998, edited by G. Giorgi and F. Rossi, Pitagora Editrice, Bologna (1999) 271-281. | Zbl 0989.47056

[16] G.J. Minty, On the generalization of a direct method of the calculus of variations. Bull. Amer. Math. Soc. 73 (1967) 314-321. | Zbl 0157.19103

[17] B. Mordukhovich, Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis. Trans. Amer. Math. Soc. 343 (1994) 609-657. | Zbl 0826.49008

[18] A.M. Rubinov, Abstract Convexity and Global Optimization, Kluwer, Dordrecht (2000). | MR 1834382 | Zbl 0985.90074

[19] H.H. Schaefer, Topological Vector Spaces. The MacMillan Company, New York, London, (1966). | MR 193469 | Zbl 0141.30503

[20] G. Stampacchia, Formes bilinéaires coercitives sur les ensembles convexes. C. R. Acad. Sci. Paris (Groupe 1) 258 (1960) 4413-4416. | Zbl 0124.06401

[21] P.T. Thach and M. Kojima, A generalized convexity and variational inequality for quasi-convex minimization. SIAM J. Optim. 6 (1996) 212-226. | Zbl 0841.49019

[22] X.Q. Yang, Generalized convex functions and vector variational inequalities. J. Optim. Theory Appl. 79 (1993) 563-580. | Zbl 0797.90085

[23] A. Zaffaroni, Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42 (2003) 1071-1086. | Zbl 1046.90084