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)
Some remarks on the Minty vector variational principle. (English) Zbl 1152.49007
Summary: In scalar optimization it is well known that a solution of a Minty variational inequality of differential type is a solution of the related optimization problem. This relation is known as “Minty variational principle.” In the vector case, the links between Minty variational inequalities and vector optimization problems were investigated in [{\it F. Giannessi}, On Minty variational principle, in: New Trends in Mathematical Programming, Dordrecht: Kluwer Academic Publishers. Appl. Optim. 13, 93--99 (1998; Zbl 0909.90253)] and subsequently in [{\it X. M. Yang, X. Q. Yang} and {\it K. L. Teo}, J. Optim. Theory Appl. 121, No. 1, 193--201 (2004; Zbl 1140.90492)]. In these papers, in the particular case of a differentiable objective function $f$ taking values in $\Bbb R^m$ and a Pareto ordering cone, it has been shown that the vector Minty variational principle holds for pseudoconvex functions. In this paper we extend such results to the case of an arbitrary ordering cone and a nondifferentiable objective function, distinguishing two different kinds of solutions of a vector optimization problem, namely ideal (or absolute) efficient points and weakly efficient points. Further, we point out that in the vector case, the Minty variational principle cannot be extended to quasiconvex functions.

49J40Variational methods including variational inequalities
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
Full Text: DOI
[1] Benoist, J.; Borwein, J. M.; Popovici, N.: A characterization of quasiconvex vector-valued functions, Proc. amer. Math. soc. 131, No. 4, 1109-1113 (2003) · Zbl 1024.26020 · doi:10.1090/S0002-9939-02-06761-8
[2] Cambini, R.: Some new classes of generalized convex vector valued functions, Optimization 36, 11-24 (1996) · Zbl 0883.26012 · doi:10.1080/02331939608844161
[3] Chabrillac, Y.; Crouzeix, J. -P.: Continuity and differentiability properties of monotone real functions of several variables, Math. prog. Study 30, 1-16 (1987) · Zbl 0611.26008
[4] Crespi, G. P.; Ginchev, I.; Rocca, M.: Minty variational inequalities, increase along rays property and optimization, J. optim. Theory appl. 123, 479-496 (2004) · Zbl 1059.49010 · doi:10.1007/s10957-004-5719-y
[5] Crespi, G. P.; Ginchev, I.; Rocca, M.: Existence of solutions and star-shapedness in minty variational inequalities, J. global optim. 32, 485-494 (2005) · Zbl 1097.49007 · doi:10.1007/s10898-003-2685-0
[6] Crespi, G. P.; Ginchev, I.; Rocca, M.: Minty vector variational inequality, efficiency and proper efficiency, Vietnam J. Math. 32, 95-107 (2004) · Zbl 1056.49009
[7] Crespi, G. P.; Ginchev, I.; Rocca, M.: Variational inequalities in vector optimization, Variational analysis and applications (2004) · Zbl 1059.49010
[8] Crespi, G. P.; Ginchev, I.; Rocca, M.: A note on minty type vector variational inequalities, RAIRO oper. Res. 39, No. 4, 253-273 (2005) · Zbl 1145.49002 · doi:10.1051/ro:2006005 · numdam:RO_2005__39_4_253_0
[9] Crespi, G. P.; Ginchev, I.; Rocca, M.: Increasing-along-rays property for vector functions, J. nonlinear convex anal. 7, No. 1, 39-50 (2006) · Zbl 1149.49008
[10] Crespi, G. P.; Ginchev, I.; Rocca, M.: Points of efficiency in vector optimization with increasing along rays property and minty variational inequalities, Generalized convexity and related topics (2007) · Zbl 1132.49008
[11] Demyanov, V. F.; Rubinov, A. M.: Constructive nonsmooth analysis, Approximation and optimization 7 (1995) · Zbl 0887.49014
[12] Giannessi, F.: On minty variational principle, New trends in mathematical programming, 93-99 (1997) · Zbl 0909.90253
[13] Ginchev, I.; Hoffmann, A.: Approximation of set-valued functions by single-valued one, Discuss. math. Differ. incl. Control optim. 22, 33-66 (2002) · Zbl 1039.90051
[14] Jameson, G.: Ordered linear spaces, Lecture notes in math. 141 (1970) · Zbl 0196.13401
[15] , Nonconvex optim. Appl. 76 (2005)
[16] Hiriart-Hurruty, J. -B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. methods oper. Res. 4, 79-97 (1979) · Zbl 0409.90086 · doi:10.1287/moor.4.1.79
[17] Luc, D. T.: Theory of vector optimization, Lecture notes in econom. And math. Systems 319 (1989)
[18] Minty, G. J.: On the generalization of a direct method of the calculus of variations, Bull. amer. Math. soc. 73, 314-321 (1967) · Zbl 0157.19103 · doi:10.1090/S0002-9904-1967-11732-4
[19] Taylor, A. E.; Lay, D. C.: Introduction to functional analysis, (1980) · Zbl 0501.46003
[20] Yang, X. M.; Yang, X. Q.; Teo, K. L.: Some remarks on the minty vector variational inequality, J. optim. Theory appl. 121, 193-201 (2004) · Zbl 1140.90492 · doi:10.1023/B:JOTA.0000026137.18526.7a