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1.
Convexlike and concavelike conditions in alternative,minimax, and minimization theorems 总被引:7,自引:0,他引:7
S. Paeck 《Journal of Optimization Theory and Applications》1992,74(2):317-332
Convexlike and concavelike conditions are of interest for extensions of the Von Neumann minimax theorem. Since the beginning of the 80's, these conditions also play a certain role in deriving generalized alternative theorems of the Gordan, Motzkin, and Farkas type and Lagrange multiplier results for constrained minimization problems.In this paper, we study various known convexlike conditions for vector-valued functions on a set and investigate convexlike and concavelike conditions for real-valued functions on a product setC×D, where we are mainly interested in the relationships between these conditions. At the end of the paper, we point out several conclusions from our results for the above-mentioned mathematical fields.The author is indebted to Dr. R. Reemtsen and Dr. V. Jeyakumar for their helpful comments during the preparations of this paper. 相似文献
2.
A function
is said to be strictly and roughly convexlike with respect to the roughness degree r > 0 (for short, strictly r-convexlike) provided that, for all x
0, x
1 D satisfying ||x
0 – x
1|| > r, there exists a ]0, 1[ such that
.The most important property of strictly r-convexlike functions is that the diameter of the set of global minimizers is not greater than r. This property is needed in another paper for obtaining the rough stability of optimal solutions to nonconvex parametric optimization problems. Moreover, if f is supposed to be lower semicontinuous, then each r-local minimizer x*, defined by
is a global minimizer of f. In this paper, necessary and sufficient conditions for a function to be strictly r-convexlike are stated. In particular, the class of strictly -convex functions is considered. 相似文献
3.
Lagrangian Duality and Cone Convexlike Functions 总被引:1,自引:0,他引:1
In this paper, we consider first the most important classes of cone convexlike vector-valued functions and give a dual characterization
for some of these classes. It turns out that these characterizations are strongly related to the closely convexlike and Ky
Fan convex bifunctions occurring within minimax problems. Applying the Lagrangian perturbation approach, we show that some
of these classes of cone convexlike vector-valued functions show up naturally in verifying strong Lagrangian duality for finite-dimensional
optimization problems. This is achieved by extending classical convexity results for biconjugate functions to the class of
so-called almost convex functions. In particular, for a general class of finite-dimensional optimization problems, strong
Lagrangian duality holds if some vector-valued function related to this optimization problem is closely K-convexlike and satisfies some additional regularity assumptions. For K a full-dimensional convex cone, it turns out that the conditions for strong Lagrangian duality simplify. Finally, we compare
the results obtained by the Lagrangian perturbation approach worked out in this paper with the results achieved by the so-called
image space approach initiated by Giannessi. 相似文献
4.
Lagrangian Duality for Minimization of Nonconvex Multifunctions 总被引:10,自引:0,他引:10
Two alternative type theorems for nearly convexlike or * quasiconvex multifunctions are presented. They are used to derive Lagrangian conditions and duality results for vector optimization problems when the objectives and the constraints are nearly convexlike or * quasiconvex multifunctions. 相似文献
5.
T. Rapcsák 《Journal of Optimization Theory and Applications》2005,127(1):177-191
The Fenchel problem of level sets in the smooth case is solved by deducing a new and nice geometric necessary and sufficient
condition for the existence of a smooth convex function with the level sets of a given smooth pseudoconvex function. The main
theorem is based on a general differential geometric tool, the space of paths defined on smooth manifolds. This approach provides
a complete geometric characterization of a new subclass of pseudoconvex functions originating from analytical mechanics and
a new view on convexlike and generalized convexlike mappings in image analysis.
This paper is dedicated to the memory of Guido Stampacchia.
The author thanks K. Balla for assistance in solving the system of differential equations of Example 2.1 and for helpful remarks.
This research was supported in part by the Hungarian Scientific Research Fund, Grants OTKA-T043276 and OTKA-T043241, and by
CNR, Rome, Italy.
An erratum to this article is available at . 相似文献
6.
Aparna Mehra 《Journal of Mathematical Analysis and Applications》2002,276(2):815-832
In this paper, we establish a scalarization theorem and a Lagrange multiplier theorem for super efficiency in vector optimization problem involving nearly convexlike set-valued maps. A dual is proposed and duality results are obtained in terms of super efficient solutions. A new type of saddle point, called super saddle point, of an appropriate set-valued Lagrangian map is introduced and is used to characterize super efficiency. 相似文献
7.
周志昂 《数学的实践与认识》2012,42(1):247-250
在局部凸空间中锥弱似凸集值映射的假设下,集值优化问题Borwein真有效解与Benson真有效解的等价性被获得.为了说明结果,一些例子被给出. 相似文献
8.
Invex-convexlike functions and duality 总被引:4,自引:0,他引:4
P. Q. Khanh 《Journal of Optimization Theory and Applications》1995,87(1):141-165
We define a class of invex-convexlike functions, which contains all convex, pseudoconvex, invex, and convexlike functions, and prove that the Kuhn-Tucker sufficient optimality condition and the Wolfe duality hold for problems involving such functions. Applications in control theory are given.The author is grateful to Professor W. Stadler and the referees for many valuable remarks and suggestions, which have enabled him to improve considerably the paper. 相似文献
9.
In this paper, we introduce a definition of generalized convexlike functions (preconvexlike functions). Then, under the weakened convexity, we study vector optimization problems in Hausdorff topological linear spaces. We establish some generalized Motzkin theorems of the alternative. By use of these theorems of the alternative, we obtain some Lagrangian multiplier theorems. A saddle-point theorem and a scalarization theorem are also derived.Communicated by F. GiannessiThe author thank Ginndomenico Mastrocni for helpful and useful comments. 相似文献
10.
A. M. Vinogradov 《Acta Appl Math》1984,2(1):21-78
Starting with Lie's classical theory, we carefully explain the basic notions of the higher symmetries theory for arbitrary systems of partial differential equations as well as the necessary calculation procedures. Roughly speaking, we explain what analogs of higher KdV equations are for an arbitrary system of partial differential equations and also how one can find and use them. The cohomological nature of conservation laws is shown and some basic results are exposed which allow one to calculate, in principle, all conservation laws for a given system of partial differential equations. In particular, it is shown that symmetry and conservation law are, in some sense, the dual conceptions which coincides in the self-dual case, namely, for Euler-Lagrange equations. Training examples are also given.Translated from the Russian by B. A. Kuperschmidt. 相似文献