Some new classes of generalized concave vector-valued functions

“…It has been shown in [4,5] that, under certain regularity assumptions on f , the so-called "Minty variational principle" holds. If x * is a solution of VI (1), then it is also a solution of the minimization problem min f (x), x ∈ K. (2) Further, under quasiconvexity assumption on f , every solution of problem (2) is a solution of the VI (1). In [12] the Minty vector variational inequality (of differential type) has been introduced and its solutions related to those of a vector optimization problem (VOP), when f : X → R m and R m is ordered by R m + .…”

Section: Introductionmentioning

confidence: 98%

Some remarks on the Minty vector variational principle

Crespi¹,

Ginchev

Rocca

2008

Journal of Mathematical Analysis and Applications

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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, Kluwer Academic, Dordrecht, 1997, pp. 93-99] and subsequently in [X.M. Yang, X.Q. Yang, K.L. Teo, Some remarks on the Minty vector variational inequality, J. Optim. Theory Appl. 121 (2004) 193-201]. In these papers, in the particular case of a differentiable objective function f taking values in 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.

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“…These functions are analyzed in detail in [14]. Other vector extensions of convexity are studied in [2], [3] and [8]. In the sequel we will use some elementary properties of K-convex functions, which we present in the following proposition; the proof may be found in almost any vector optimization book (see, for instance, [14]).…”

Section: Review Of Elementary Resultsmentioning

confidence: 98%

On First Order Optimality Conditions for Vector Optimization

Drummond

Iusem

Svaiter

2003

Acta Mathematicae Applicatae Sinica, English Series, English Se

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We develop first order optimality conditions for constrained vector optimization. The partial orders for the objective and the constraints are induced by closed and convex cones with nonempty interior. After presenting some well known existence results for these problems, based on a scalarization approach, we establish necessity of the optimality conditions under a Slater-like constraint qualification, and then sufficiency for the K-convex case. We present two alternative sets of optimality conditions, with the same properties in connection with necessity and sufficiency, but which are different with respect to the dimension of the spaces to which the dual multipliers belong. We introduce a duality scheme, with a point-to-set dual objective, for which strong duality holds. Some examples and open problems for future research are also presented.

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Some new classes of generalized concave vector-valued functions

Cited by 42 publications

References 3 publications

Symmetric duality in multiobjective programming involving generalized cone-invex functions

Symmetric duality in multiobjective programming involving generalized cone-invex functions

Some remarks on the Minty vector variational principle

On First Order Optimality Conditions for Vector Optimization

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