zbMATH — the first resource for mathematics

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 [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 [X. M. Yang, X. Q. Yang and 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 \(\mathbb R^m\) 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.

49J40 Variational inequalities
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
Full Text: DOI
[1] Benoist, J.; Borwein, J.M.; Popovici, N., A characterization of quasiconvex vector-valued functions, Proc. amer. math. soc., 131, 4, 1109-1113, (2003), (electronic) · Zbl 1024.26020
[2] Cambini, R., Some new classes of generalized convex vector valued functions, Optimization, 36, 11-24, (1996) · Zbl 0883.26012
[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
[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
[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, () · Zbl 1132.90015
[8] Crespi, G.P.; Ginchev, I.; Rocca, M., A note on minty type vector variational inequalities, RAIRO oper. res., 39, 4, 253-273, (2005) · Zbl 1145.49002
[9] Crespi, G.P.; Ginchev, I.; Rocca, M., Increasing-along-rays property for vector functions, J. nonlinear convex anal., 7, 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, () · Zbl 1059.49010
[11] Demyanov, V.F.; Rubinov, A.M., Constructive nonsmooth analysis, Approximation and optimization, vol. 7, (1995), Peter Lang Frankfurt am Main · Zbl 0887.49014
[12] Giannessi, F., On minty variational principle, (), 93-99 · 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., vol. 141, (1970), Springer Berlin · Zbl 0196.13401
[15] ()
[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
[17] Luc, D.T., Theory of vector optimization, Lecture notes in econom. and math. systems, vol. 319, (1989), Springer-Verlag Berlin
[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
[19] Taylor, A.E.; Lay, D.C., Introduction to functional analysis, (1980), John Wiley & Sons New York-Chichester-Brisbane
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.