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1.
V. V. Makeev 《Journal of Mathematical Sciences》2007,140(4):529-534
A convex figure K ⊂ ℝ2 is a compact convex set with nonempty interior, and αK is a homothetic image of K with coefficient α ∈ ℝ. It is proved that
for any two convex figures K1, K2 ⊂ ℝ2 there is an affine transformation T of the plane such that K1 ⊂ T(K2) ⊂ 2.7K1. Bibliography: 2 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 58–66. 相似文献
2.
Satoshi Suzuki 《Journal of Global Optimization》2010,47(2):273-285
Dual characterizations of the containment of a convex set, defined by infinite quasiconvex constraints, in an evenly convex
set, and in a reverse convex set, defined by infinite quasiconvex constraints, are provided. Notions of quasiconjugate for
quasiconvex functions, λ-quasiconjugate and λ-semiconjugate, play important roles to derive the characterizations of the set
containments. 相似文献
3.
The relation of time indexed formulations of nonpreemptive single machine scheduling problems to the node packing problem
is established and then used to provide simple and intuitive alternate proofs of validity and maximality for previously known
results on the facial structure of the scheduling problem. Previous work on the facial structure has focused on describing
the convex hull of the set of feasible partial schedules, schedules in which not all jobs have to be started. The equivalence
between the characteristic vectors of this set and those of the set of feasible node packings in a graph whose structure is
determined by the parameters of the scheduling problem is established. The main contribution of this paper is to show that
the facet inducing inequalities for the convex hull of the set of feasible partial schedules that have integral coefficients
and right hand side 1 or 2 are the maximal clique inequalities and the maximally and sequentially lifted 5-hole inequalities
of the convex hull of the set of feasible node packings in this graph respectively.
Received: September 10, 2000 / Accepted: April 20, 2002 Published online: September 27, 2002
Key words. scheduling – node packing – polyhedral methods – facet defining graphs – lifted valid inequalities – facet inducing inequalities} 相似文献
4.
We consider a quadratic cut method based on analytic centers for two cases of convex quadratic feasibility problems. In the
first case, the convex set is defined by a finite yet large number, N, of convex quadratic inequalities. We extend quadratic cut algorithm of Luo and Sun [3] for solving such problems by placing
or translating the quadratic cuts directly through the current approximate center. We show that, in terms of total number
of addition and translation of cuts, our algorithm has the same polynomial worst case complexity as theirs [3]. However, the
total number of steps, where steps consist of (damped) Newton steps, function evaluations and arithmetic operations, required
to update from one approximate center to another is , where ε is the radius of the largest ball contained in the feasible set. In the second case, the convex set is defined by
an infinite number of certain strongly convex quadratic inequalities. We adapt the same quadratic cut method for the first
case to the second one. We show that in this case the quadratic cut algorithm is a fully polynomial approximation scheme.
Furthermore, we show that, at each iteration, k, the total number steps (as described above) required to update from one approximate center to another is at most , with ε as defined above.
Received: April 2000 / Accepted: June 2002 Published online: September 5, 2002
Key words. convex quadratic feasibility problem – interior-point methods – analytic center – quadratic cuts – potential function 相似文献
5.
I. Bárány 《Discrete and Computational Geometry》1995,13(1):279-295
It is proved here that, asn→∞, almost all convex (1/n)ℤ2-lattice polygons lying in the square [−1, 1]2 are very close to a fixed convex set.
This research was partially supported by Hungarian Science Foundation Grants 1907 and 1909. 相似文献
6.
Kalpana Dahiya Surjeet Kaur Suneja Vanita Verma 《Computational Optimization and Applications》2007,36(1):67-82
In this paper a minimization problem with convex objective function subject to a separable convex inequality constraint “≤”
and bounded variables (box constraints) is considered. We propose an iterative algorithm for solving this problem based on
line search and convergence of this algorithm is proved. At each iteration, a separable convex programming problem with the
same constraint set is solved using Karush-Kuhn-Tucker conditions. Convex minimization problems subject to linear equality/
linear inequality “≥” constraint and bounds on the variables are also considered. Numerical illustration is included in support
of theory. 相似文献
7.
8.
V. V. Makeev 《Journal of Mathematical Sciences》1998,91(6):3518-3525
Five theorems on polygons and polytopes inscribed in (or circumscribed about) a convex compact set in the plane or space are
proved by topological methods. In particular, it is proved that for every interior point O of a convex compact set in ℝ3, there exists a two-dimensional section through O circumscribed about an affine image of a regular octagon. It is also proved
that every compact convex set in ℝ3 (except the cases listed below) is circumscribed about an affine image of a cube-octahedron (the convex hull of the midpoints
of the edges of a cube). Possible exceptions are provided by the bodies containing a parallelogram P and contained in a cylinder
with directrix P. Bibliography: 29 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 286–298.
Translated by B. M. Bekker. 相似文献
9.
A finite set of points, in general position in the plane, is almost convex if every triple determines a triangle with at most one point in its interior. For every ℓ ≥ 3, we determine the maximum size
of an almost convex set that does not contain the vertex set of an empty convex ℓ-gon.
Partially supported by grants T043631 and NK67867 of the Hungarian NFSR (OTKA). 相似文献
10.
Herbert Busemann 《Annali di Matematica Pura ed Applicata》1961,55(1):171-189
Summary It is shown that for 1≤a≤n an open convex set in n-dimensional affine space can be provided with a complete a-dimensional
area for which the a-flats minimize area, and that any such area can be extended to an area with the same propertiés in an
open convex set in (n+1)—space.
To Enrico Bompiani on his scientific Jubilee.
This work was supported by a grant from the National Science Foundation. 相似文献
11.
We exhibit a collection of extreme points of the family of normalized convex mappings of the open unit ball of ℂ
n
forn≥2. These extreme points are defined in terms of the extreme points of a closed ball in the Banach space of homogeneous polynomials
of degree 2 in ℂ
n−1, which are fully classified. Two examples are given to show that there are more convex mappings than those contained in the
closed convex hull of the set of extreme points here exhibited. 相似文献
12.
A. Auslender 《Mathematical Programming》2000,88(1):45-59
In this paper we consider an ordinary convex program with no qualification conditions (such as Slater’s condition) and for
which the optimal set is neither required to be compact, nor to be equal to the sum of a compact set and a linear space. It
is supposed only that the infimum α is finite. A very wide class of convex functions is exhibited for which the optimum is
always attained and α is equal to the supremum of the ordinary dual program. Additional results concerning the existence of
optimal solutions in the non convex case are also given.
Received: February 1, 1999 / Accepted: December 15, 1999?Published online February 23, 2000 相似文献
13.
In this paper, we investigate the problems of robust delay-dependent ℒ2 gain analysis and feedback control synthesis for a class of nominally-linear switched discrete-time systems with time-varying
delays, bounded nonlinearities and real convex bounded parametric uncertainties in all system matrices under arbitrary switching
sequences. We develop new criteria for such class of switched systems based on the constructive use of an appropriate switched
Lyapunov-Krasovskii functional coupled with Finsler’s Lemma and a free-weighting parameter matrix. We establish an LMI characterization of delay-dependent conditions under which the
nonlinear switched delay system is robustly asymptotically stable with an ℒ2-gain smaller than a prescribed constant level. Switched feedback schemes, based on state measurements, output measurements
or by using dynamic output feedback, are designed to guarantee that the corresponding switched closed-loop system enjoys the
delay-dependent asymptotic stability with an ℒ2 gain smaller than a prescribed constant level. All the developed results are expressed in terms of convex optimization over
LMIs and tested on representative examples. 相似文献
14.
Every relatively convex-compact convex subset of a locally convex space is contained in a Banach disc. Moreover, an upper
bound for the class of sets which are contained in a Banach disc is presented. If the topological dual E′ of a locally convex space E is the σ(E′,E)-closure of the union of countably many σ(E′,E)-relatively countably compacts sets, then every weakly (relatively) convex-compact set is weakly (relatively) compact. 相似文献
15.
A geometric permutation induced by a transversal line of a finite family of disjoint convex sets in ℝd is the order in which the transversal meets the members of the family. It is known that the maximal number of geometric permutations
in families of n disjoint translates of a convex set in ℝ3 is 3. We prove that for d ≥ 3 the maximal number of geometric permutations for such families in ℝd is Ω(n). 相似文献
16.
H. Mäurer 《Monatshefte für Mathematik》1999,128(1):7-21
A bijection of the compact convex set with is called a reflection of K, if σ maps convex subsets of K into convex subsets. Conditions are stated unter which the existence of few reflections imply that is an ellipse.
Received 4 March, 1998 相似文献
17.
A. V. Chernov 《Differential Equations》2012,48(4):586-595
We consider a controlled functional-operator equation that is a convenient form for describing of controlled initial-boundary
value problems. For this equation, considered in a Banach ideal space, we define the set Ω of global solvability as the set
of all admissible controls for which the equation has a global solution. We show that Ω is convex under the conditions imposed
on the right-hand side of the equation. For each control in a given segment in Ω, we obtain a two-sided pointwise estimate
for the corresponding solution under the abovementioned assumptions. We prove the theorem on the convex continuous dependence
of the solution on the parameter specifying the displacement along the segment. 相似文献
18.
Yu. É. Linke 《Mathematical Notes》1998,63(2):183-189
The existence of separately continuous selectors of a separately lower and upper semicontinuous many-valued multivariate map
is proved, provided its domain is a compact set and its range is a convex closed subset of a metrizable convex compact set.
Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 209–216, February, 1998.
This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00264. 相似文献
19.
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
20.
Yves Benoist 《Inventiones Mathematicae》2006,164(2):249-278
Divisible convex sets IV: Boundary structure in dimension 3
Let Ω be an indecomposable properly convex open subset of the real projective 3-space which is divisible i.e. for which there exists a torsion free discrete group Γ of projective transformations preserving Ω such that the quotient
M := Γ\Ω is compact. We study the structure of M and of ∂Ω, when Ω is not strictly convex:
The union of the properly embedded triangles in Ω projects in M onto an union of finitely many disjoint tori and Klein bottles which induces an atoroidal decomposition of M.
Every non extremal point of ∂Ω is on an edge of a unique properly embedded triangle in Ω and the set of vertices of these
triangles is dense in the boundary of Ω (see Figs. 1 to 4).
Moreover, we construct examples of such divisible convex open sets Ω.
相似文献