共查询到20条相似文献,搜索用时 31 毫秒
1.
2.
主要研究了两类近似凸集的关系和性质.首先,举例说明两类近似凸集没有相互包含关系.其次,在近似凸集(nearly convex)条件下,证明了在一定条件下函数上图是近似凸集与凸集的等价关系.同时,考虑了近似凸函数与函数上图是近似凸集的等价刻画、近似凸函数与函数水平集是近似凸集的必要性,并用例子说明近似凸函数与函数水平集是... 相似文献
3.
In this paper, we study properties of general closed convex sets that determine the closedness and polyhedrality of the convex hull of integer points contained in it. We first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed. This leads to useful results for special classes of convex sets such as pointed cones, strictly convex sets, and sets containing integer points in their interior. We then present a sufficient condition for the convex hull of integer points in general convex sets to be a polyhedron. This result generalizes the well-known result due to Meyer (Math Program 7:223–225, 1974). Under a simple technical assumption, we show that these sufficient conditions are also necessary for the convex hull of integer points contained in general convex sets to be a polyhedron. 相似文献
4.
Patrick N. Dowling 《Journal of Mathematical Analysis and Applications》2003,288(2):540-543
The ψ-direct sum of Banach spaces is strictly convex (respectively, uniformly convex, locally uniformly convex, uniformly convex in every direction) if each of the Banach spaces are strictly convex (respectively, uniformly convex, locally uniformly convex, uniformly convex in every direction) and the corresponding ψ-norm is strictly convex. 相似文献
5.
Generalized polyhedral convex sets, generalized polyhedral convex functions on locally convex Hausdorff topological vector spaces, and the related constructions such as sum of sets, sum of functions, directional derivative, infimal convolution, normal cone, conjugate function, subdifferential are studied thoroughly in this paper. Among other things, we show how a generalized polyhedral convex set can be characterized through the finiteness of the number of its faces. In addition, it is proved that the infimal convolution of a generalized polyhedral convex function and a polyhedral convex function is a polyhedral convex function. The obtained results can be applied to scalar optimization problems described by generalized polyhedral convex sets and generalized polyhedral convex functions. 相似文献
6.
Walter Roth has investigated certain equivalence relations on locally convex cones in [W. Roth, Locally convex quotient cones, J. Convex Anal. 18, No. 4, 903–913 (2011)] which give rise to the definition of a locally convex quotient cone. In this paper, we investigate some special equivalence relations on a locally convex lattice cone by which the locally convex quotient cone becomes a lattice. In the case of a locally convex solid Riesz space, this reduces to the known concept of locally convex solid quotient Riesz space. We prove that the strict inductive limit of locally convex lattice cones is a locally convex lattice cone. We also study the concept of locally convex complete quotient lattice cones. 相似文献
7.
Strong restricted-orientation convexity is a generalization of standard convexity. We explore the properties of strongly convex sets in multidimensional Euclidean space and identify major properties of standard convex sets that also hold for strong convexity. We characterize strongly convex flats and halfspaces, and establish the strong convexity of the affine hull of a strongly convex set. We then show that, for every point in the boundary of a strongly convex set, there is a supporting strongly convex hyperplane through it. Finally, we show that a closed set with nonempty interior is strongly convex if and only if it is the intersection of strongly convex halfspaces; we state a condition under which this result extends to sets with empty interior. 相似文献
8.
In this article the existence of the convex extension of convex set valued map is considered. Conditions are obtained, based on the notion of the derivative of set valued maps, which guarantee the existence of convex extension. The conditions are given, when the convex set valued map has no convex extension. The convex set valued map is specified, which is the maximal convex extension of the given convex set valued map and includes all other convex extensions. The connection between Lipschitz continuity and existence of convex extension of the given convex set valued map is studied. 相似文献
9.
J. Brinkhuis 《Journal of Optimization Theory and Applications》2009,143(3):439-453
An attempt is made to justify results from Convex Analysis by means of one method. Duality results, such as the Fenchel-Moreau
theorem for convex functions, and formulas of convex calculus, such as the Moreau-Rockafellar formula for the subgradient
of the sum of sublinear functions, are considered. All duality operators, all duality theorems, all standard binary operations,
and all formulas of convex calculus are enumerated. The method consists of three automatic steps: first translation from the
given setting to that of convex cones, then application of the standard operations and facts (the calculi) for convex cones,
finally translation back to the original setting. The advantage is that the calculi are much simpler for convex cones than
for other types of convex objects, such as convex sets, convex functions and sublinear functions. Exclusion of improper convex
objects is not necessary, and recession directions are allowed as points of convex objects. The method can also be applied
beyond the enumeration of the calculi. 相似文献
10.
《Optimization》2012,61(4):541-560
This paper concerns a closedness condition called (CC) involving a convex function and a convex constrained system. This type of condition has played an important role in the study of convex optimization problems. Our aim is to establish several characterizations of this condition and to apply them to study problems of minimizing a DC function under a cone-convex constraint and a set constraint. First, we establish several so-called ‘Toland–Fenchel–Lagrange’ duality theorems. As consequences, various versions of generalized Farkas lemmas in dual forms for systems involving convex and DC functions are derived. Then, we establish optimality conditions for DC problem under convex constraints. Optimality conditions for convex problems and problems of maximizing a convex function under convex constraints are given as well. Most of the results are established under the (CC) condition. This article serves as a link between several corresponding known ones published recently for DC programs and for convex programs. 相似文献
11.
Marek Cezary Zdun 《Journal of Difference Equations and Applications》2018,24(5):773-783
AbstractA function f is said to be iteratively convex if f possesses convex iterative roots of all orders. We give several constructions of iteratively convex diffeomorphisms and explain the phenomenon that the non-existence of convex iterative roots is a typical property of convex functions. We show also that a slight perturbation of iteratively convex functions causes loss of iterative convexity. However, every convex function can be approximate by iteratively convex functions. 相似文献
12.
In this paper some concepts of convex analysis are extended in an intrinsic way from the Euclidean space to the sphere. In particular, relations between convex sets in the sphere and pointed convex cones are presented. Several characterizations of the usual projection onto a Euclidean convex set are extended to the sphere and an extension of Moreau’s theorem for projection onto a pointed convex cone is exhibited. 相似文献
13.
交替最小化算法(简称AMA)最早由[SIAM J.Control Optim.,1991,29(1):119-138]提出,并能用于求解强凸函数与凸函数和的极小值问题.本文直接利用AMA算法来求解强凸函数与弱凸函数和的极小值问题.在强凸函数的模大于弱凸函数的模的假设下,我们证明了AMA生成的点列全局收敛到优化问题的解,并且若该优化问题中的某个函数是光滑函数时,AMA所生成的点列的收敛率是线性的. 相似文献
14.
We examine a notion of generalized convex set-valued mapping, extending the notions of a convex relation and a convex process. Under general conditions, we establish duality results for composite set-valued mappings and for convex programming problems involving convex set-valued mappings. We also present applications to the study of economic dynamical systems, by obtaining the characteristics of optimal paths generated by convex processes, and to optimization problems of a certain class of positively homogeneous increasing functions. 相似文献
15.
16.
Mohsen Alimohammady Yeol Je Cho Vahid Dadashi Mehdi Roohi 《Applied Mathematics Letters》2011,24(8):1289-1294
This paper deals with some basic notions of convex analysis and convex optimization via convex semi-closed functions. A decoupling-type result and also a sandwich theorem are proved. As a consequence of the sandwich theorem, we get a convex sub-differential sum rule and two separation results. Finally, the derived convex sub-differential sum rule is applied to solving the convex programming problem. 相似文献
17.
We define a convex extension of a lower semi-continuous function to be a convex function that is identical to the given function
over a pre-specified subset of its domain. Convex extensions are not necessarily constructible or unique. We identify conditions
under which a convex extension can be constructed. When multiple convex extensions exist, we characterize the tightest convex
extension in a well-defined sense. Using the notion of a generating set, we establish conditions under which the tightest
convex extension is the convex envelope. Then, we employ convex extensions to develop a constructive technique for deriving
convex envelopes of nonlinear functions. Finally, using the theory of convex extensions we characterize the precise gaps exhibited
by various underestimators of $x/y$ over a rectangle and prove that the extensions theory provides convex relaxations that
are much tighter than the relaxation provided by the classical outer-linearization of bilinear terms.
Received: December 2000 / Accepted: May 2002 Published online: September 5, 2002
RID="*"
ID="*" The research was funded in part by a Computational Science and Engineering Fellowship to M.T., and NSF CAREER award
(DMI 95-02722) and NSF/Lucent Technologies Industrial Ecology Fellowship (NSF award BES 98-73586) to N.V.S.
Key words. convex hulls and envelopes – multilinear functions – disjunctive programming – global optimization 相似文献
18.
This paper considers planar location problems with rectilinear distance and barriers where the objective function is any convex, nondecreasing function of distance. Such problems have a non-convex feasible region and a nonconvex objective function. Based on an equivalent problem with modified barriers, derived in a companion paper [3], the non convex feasible set is partitioned into a network and rectangular cells. The rectangular cells are further partitioned into a polynomial number of convex subcells, called convex domains, on which the distance function, and hence the objective function, is convex. Then the problem is solved over the network and convex domains for an optimal solution. Bounds are given that reduce the number of convex domains to be examined. The number of convex domains is bounded above by a polynomial in the size of the problem. 相似文献
19.
S. Schaible 《Journal of Optimization Theory and Applications》1976,19(2):347-352
Optimization problems are considered where ratios of functionals of several different types (convex/concave, concave/convex, convex/convex, and concave/concave) are to be minimized over a convex set. All these problems are related to the minimization of a quasi-convex functional or a quasi-concave functional. 相似文献
20.
Lars Michael Hoffmann 《Acta Appl Math》2009,105(2):141-156
Translative integral formulas for curvature measures of convex bodies were obtained by Schneider and Weil by introducing mixed
measures of convex bodies. These results can be extended to arbitrary closed convex sets since mixed measures are locally
defined. Furthermore, iterated versions of these formulas due to Weil were used by Fallert to introduce quermass densities
for (non-stationary and non-isotropic) Poisson processes of convex bodies and respective Boolean models. In the present paper,
we first compute the special form of mixed measures of convex cylinders and prove a translative integral formula for them.
After adapting some results for mixed measures of convex bodies to this setting we then use this integral formula to obtain
quermass densities for (non-stationary and non-isotropic) Poisson processes of convex cylinders. Furthermore, quermass densities
of Boolean models of convex cylinders are expressed in terms of mixed densities of the underlying Poisson process generalizing
classical formulas by Davy and recent results by Spiess and Spodarev.
相似文献