共查询到10条相似文献,搜索用时 109 毫秒
1.
Phil Diamond Peter Kloeden Alexander Rubinov Alexander Vladimirov 《Set-Valued Analysis》1997,5(3):267-289
Along with the Hausdorff metric, we consider two other metrics on the space of convex sets, namely, the metric induced by the Demyanov difference of convex sets and the Bartels–Pallaschke metric. We describe the hierarchy of these three metrics and of the corresponding norms in the space of differences of sublinear functions. The completeness of corresponding metric spaces is demonstrated. Conditions of differentiability of convex-valued maps of one variable with respect to these metrics are proved for some special cases. Applications to the theory of convex fuzzy sets are given. 相似文献
2.
Yan Gao 《Journal of Applied Mathematics and Computing》2001,8(2):381-397
The notion of difference for two convex compact sets inR n , proposed by Rubinovet al, is generalized toR m×n . A formula of the difference for the two sets, which are convex hulls of a finite number of points, is developed. In the light of this difference, the relation between Clarke generalized Jacobian and quasidifferential, in the sense of Demyanov and Rubinov, for a nonsnooth function, is established. Based on the relation, the method of estimating Clarke generalized Jacobian via quasidifferential for a certain class of functions, is presented. 相似文献
3.
Representative of Quasidifferentials and Its Formula for a Quasidifferentiable Function 总被引:1,自引:0,他引:1
Yan Gao 《Set-Valued Analysis》2005,13(4):323-336
The quasidifferential of a quasidifferentiable function in the sense of Demyanov and Rubinov is not uniquely defined. Xia
proposed the notion of the kernelled quasidifferential, which is expected to be a representative for the equivalent class
of quasidifferentials. In the 2-dimensional case, the existence of the kernelled quasidifferential was shown. In this paper,
the existence of the kernelled quasidifferential in the n-dimensional space (n>2) is proved under the assumption that the Minkowski difference and the Demyanov difference of subdifferential and minus
superdifferential coincide. In particular, given a quasidifferential, the kernelled quasidifferential can be formulated. Applications
to two classes of generalized separable quasidifferentiable functions are developed.
Mathematics Subject Classifications (2000) 49J52, 54C60, 90C26.
This work was supported by Shanghai Education Committee (04EA01). 相似文献
4.
Joydeep Dutta 《TOP》2005,13(2):185-279
During the early 1960’s there was a growing realization that a large number of optimization problems which appeared in applications
involved minimization of non-differentiable functions. One of the important areas where such problems appeared was optimal
control. The subject of nonsmooth analysis arose out of the need to develop a theory to deal with the minimization of nonsmooth
functions. The first impetus in this direction came with the publication of Rockafellar’s seminal work titledConvex Analysis which was published by the Princeton University Press in 1970. It would be impossible to overstate the impact of this book
on the development of the theory and methods of optimization. It is also important to note that a large part of convex analysis
was already developed by Werner Fenchel nearly twenty years earlier and was circulated through his mimeographed lecture notes
titledConvex Cones, Sets and Functions, Princeton University, 1951. In this article we trace the dramatic development of nonsmooth analysis and its applications
to optimization in finite dimensions. Beginning with the fundamentals of convex optimization we quickly move over to the path
breaking work of Clarke which extends the domain of nonsmooth analysis from convex to locally Lipschitz functions. Clarke
was the second doctoral student of R.T. Rockafellar. We discuss the notions of Clarke directional derivative and the Clarke
generalized gradient and also the relevant calculus rules and applications to optimization. While discussing locally Lipschitz
optimization we also try to blend in the computational aspects of the theory wherever possible. This is followed by a discussion
of the geometry of sets with nonsmooth boundaries. The approach to develop the notion of the normal cone to an arbitrary set
is sequential in nature. This approach does not rely on the standard techniques of convex analysis. The move away from convexity
was pioneered by Mordukhovich and later culminated in the monographVariational Analysis by Rockafellar and Wets. The approach of Mordukhovich relied on a nonconvex separation principle called theextremal principle while that of Rockafellar and Wets relied on various convergence notions developed to suit the needs of optimization. We
then move on to a parallel development in nonsmooth optimization due to Demyanov and Rubinov called Quasidifferentiable optimization.
They study the class of directionally differentiable functions whose directional derivatives can be represented as a difference
of two sublinear functions. On other hand the directional derivative of a convex function and also the Clarke directional
derivatives are sublinear functions of the directions.
Thus it was thought that the most useful generalizations of directional derivatives must be a sublinear function of the directions.
Thus Demyanov and Rubinov made a major conceptual change in nonsmooth optimization. In this section we define the notion of
a quasidifferential which is a pair of convex compact sets. We study some calculus rules and their applications to optimality
conditions. We also study the interesting notion of Demyanov difference between two sets and their applications to optimization.
In the last section of this paper we study some second-order tools used in nonsmooth analysis and try to see their relevance
in optimization. In fact it is important to note that unlike the classical case, the second-order theory of nonsmoothness
is quite complicated in the sense that there are many approaches to it. However we have chosen to describe those approaches
which can be developed from the first order nonsmooth tools discussed here. We shall present three different approaches, highlight
the second order calculus rules and their applications to optimization. 相似文献
5.
V.F. Demyanov introduced exhausters for the study of nonsmooth functions. These are families of convex compact sets that enable one to represent the main part of the increment of a considered function in a neighborhood of the studied point as MaxMin or MinMax of linear functions. Optimality conditions were described in terms of these objects. This provided a way for constructing new algorithms for solving nondifferentiable optimization problems. Exhausters are defined not uniquely. It is obvious that the smaller an exhauster, the less are the computational expenses when working with it. Thus, the problem of reduction of an available family arises. For the first time, this problem was considered by V.A. Roshchina. She proposed conditions for minimality and described some methods of reduction in the case when these conditions are not satisfied. However, it turned out that the exhauster mapping is not continuous in the Hausdorff metrics, which leads to the problems with convergence of numerical methods. To overcome this difficulty, Demyanov proposed the notion of coexhausters. These objects enable one to represent the main part of the increment of the considered function in a neighborhood of the studied point in the form of MaxMin or MinMax of affine functions. One can define a class of functions with the continuous coexhauster mapping. Optimality conditions can be stated in terms of these objects too. But coexhausters are also defined not uniquely. The problem of reduction of coexhausters is considered in this paper for the first time. Definitions of minimality proposed by Roshchina are used. In contrast to ideas proposed in the works of Roshchina, the minimality conditions and the technique of reduction developed in this paper have a clear and transparent geometric interpretation. 相似文献
6.
A family of probability measures on the unit ball in generates a family of generalized Steiner (GS-)points for every convex compact set in . Such a “rich” family of probability measures determines a representation of a convex compact set by GS-points. In this way,
a representation of a set-valued map with convex compact images is constructed by GS-selections (which are defined by the
GS-points of its images). The properties of the GS-points allow to represent Minkowski sum, Demyanov difference and Demyanov
distance between sets in terms of their GS-points, as well as the Aumann integral of a set-valued map is represented by the
integrals of its GS-selections. Regularity properties of set-valued maps (measurability, Lipschitz continuity, bounded variation)
are reduced to the corresponding uniform properties of its GS-selections. This theory is applied to formulate regularity conditions
for the first-order of convergence of iterated set-valued quadrature formulae approximating the Aumann integral.
相似文献
7.
《Optimization》2012,61(3):179-188
Operation of the difference of pairs of convex compacta introduced by Demyanov and the related operation proposed by the authors are investigated. Using these operations the relationship between the Clarke subdifferential and quasidifferential is clarified 相似文献
8.
9.
Chun-Ling Song Zun-Quan Xia Li-Wei Zhang Shu-Yang Li 《Journal of Applied Mathematics and Computing》2007,23(1-2):353-359
A necessary and sufficient condition for Demyanov difference and Minkowski difference of compact convex subsets inR 2 being equal is given in this paper. Several examples are computed by Matlab to test our result. The necessary and sufficient condition makes us to compute Clarke subdifferential by quasidifferential for a special of Lipschitz functions. 相似文献
10.
V. Roshchina 《Journal of Optimization Theory and Applications》2008,136(2):261-273
The notions of upper and lower exhausters were introduced by Demyanov (Optimization 45:13–29, 1999). Upper and lower exhausters can be employed to study a very wide range of positively homogeneous functions, for example,
various directional derivatives of nonsmooth functions. Exhausters are not uniquely defined; hence, the problem of minimality
arises naturally. This paper describes some techniques for reducing exhausters, both in size and amount of sets. We define
also a modified convertor which provides much more flexibility in converting upper exhausters to lower ones and vice versa,
and allows us to obtain much smaller sets. 相似文献