zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Nonsmooth vector optimization problems and Minty vector variational inequalities. (English) Zbl 1198.90343
Authors’ abstract: The vector optimization problem may have a nonsmooth objective function. Therefore, we introduce the Minty vector variational inequality (Minty VVI) and the Stampacchia vector variational inequality (Stampacchia VVI) defined by means of upper Dini derivative. By using the Minty VVI, we provide a necessary and sufficient condition for a vector minimal point (v.m.p.) of a vector optimization problem for pseudoconvex functions involving Dini derivatives. We establish the relationship between the Minty VVI and the Stampacchia VVI under upper sign continuity. Some relationships among v.m.p., weak v.m.p., solutions of the Stampacchia VVI and solutions of the Minty VVI are discussed. We present also an existence result for the solutions of the weak Minty VVI and the weak Stampacchia VVI.

90C29Multi-objective programming; goal programming
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
Full Text: DOI
[1] Al-Homidan, S., Ansari, Q.H.: Generalized minty vector variational-like inequalities and vector optimization problems. J. Optim. Theory Appl. 144(1), (2010) · Zbl 1247.90227
[2] Ansari, Q.H., Yao, J.C.: On nondifferentaible and nonconvex vector optimization problems. J. Optim. Theory Appl. 106(3), 475--488 (2000) · Zbl 0970.90092 · doi:10.1023/A:1004697127040
[3] Giannessi, F.: On minty variational principle. In: Giannessi, F., Komlòsi, S., Tapcsáck, T. (eds.) New Trends in Mathematical Programming, pp. 93--99. Kluwer Academic, Dordrecht (1998) · Zbl 0909.90253
[4] Giannessi, F.: Vector Variational Inequalities and Vector Equilibria: Mathematical Theories. Kluwer Academic, Dordrecht (2000) · Zbl 0952.00009
[5] Komlòsi, S.: On the stampacchia and minty variational inequalities. In: Giorgi, G., Rossi, F. (eds.) Generalized Convexity and Optimization for Economic and Financial Decisions, pp. 231--260. Pitagora Editrice, Bologna (1999) · Zbl 0989.47055
[6] Lee, G.M.: On relations between vector variational inequality and vector optimization problem. In: Yang, X.Q., Mees, A.I., Fisher, M.E., Jennings, L.S. (eds.) Progress in Optimization, II: Contributions from Australasia, pp. 167--179. Kluwer Academic, Dordrecht (2000) · Zbl 0969.49003
[7] Lee, G.M., Kim, D.S., Kuk, H.: Existence of solutions for vector optimization problems. J. Math. Anal. Appl. 220, 90--98 (1998) · Zbl 0911.90290 · doi:10.1006/jmaa.1997.5821
[8] Lee, G.M., Kim, D.S., Lee, B.S., Yen, N.D.: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Anal.: Theory Methods Appl. 34, 745--765 (1998) · Zbl 0956.49007 · doi:10.1016/S0362-546X(97)00578-6
[9] Ruiz-Garzón, G., Osuna-Gómez, R., Rufián-Lizana, A.: Relationships between vector variational-like inequality and optimization problems. Eur. J. Oper. Res. 157, 113--119 (2004) · Zbl 1106.90060 · doi:10.1016/S0377-2217(03)00210-8
[10] Yang, X.M., Yang, X.Q., Teo, K.L.: Some remarks on the minty vector variational inequality. J. Optim. Theory Appl. 121(1), 193--201 (2004) · Zbl 1140.90492 · doi:10.1023/B:JOTA.0000026137.18526.7a
[11] Crespi, G.P., Ginchev, I., Rocca, M.: Minty variational inequalities, increase along rays property and optimization. J. Optim. Theory Appl. 123(3), 479--496 (2004) · Zbl 1059.49010 · doi:10.1007/s10957-004-5719-y
[12] Crespi, G.P., Ginchev, I., Rocca, M.: Existence of solutions and star-shapedness in minty variational inequalities. J. Glob. Optim. 32, 485--494 (2005) · Zbl 1097.49007 · doi:10.1007/s10898-003-2685-0
[13] Crespi, G.P., Ginchev, I., Rocca, M.: Some remarks on the minty vector variational principle. J. Math. Anal. Appl. 345, 165--175 (2008) · Zbl 1152.49007 · doi:10.1016/j.jmaa.2008.03.012
[14] Sach, P.H., Penot, J.-P.: Charaterizations of generalized convexities via generalized directional derivative. Numer. Funct. Anal. Optim. 19(5--6), 615--634 (1998) · Zbl 0916.49015 · doi:10.1080/01630569808816849
[15] Komlòsi, S.: Generalized monotonicity and generalized convexity. J. Optim. Theory Appl. 84(2), 361--376 (1995) · Zbl 0824.90124 · doi:10.1007/BF02192119
[16] Giorgi, G., Komlósi, S.: Dini derivatives in optimization--Part III. Riv. Mat. Sci. Econ. Soc. 18(1), 47--63 (1995) · Zbl 0884.90134
[17] Diewert, W.E.: Alternative charaterizations of six kinds of quasiconcavity in the nondifferentiable case with applications to nonsmooth programming. In: Schaible, S., Ziemba, W.T. (eds.) Generalized Concavity in Optimization and Economics, pp. 51--93. Academic Press, New York (1981) · Zbl 0539.90088
[18] Lalitha, C.S., Mehta, M.: Vector variational inequalities with cone-pseudomonotone bifunctions. Oprimization 54(3), 327--338 (2005) · Zbl 1087.90069 · doi:10.1080/02331930500100254
[19] Hadjisavvas, N.: Continuity and maximality properties of pseudomonotone operators. J. Convex Anal. 10(2), 459--469 (2003) · Zbl 1063.47041
[20] Fan, K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142, 305--310 (1961) · Zbl 0093.36701 · doi:10.1007/BF01353421