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1.
We consider the diagonal inexact proximal point iteration where f(x,r)=c
T
x+r∑exp[(A
i
x-b
i
)/r] is the exponential penalty approximation of the linear program min{c
T
x:Ax≤b}. We prove that under an appropriate choice of the sequences λ
k
, ε
k
and with some control on the residual ν
k
, for every r
k
→0+ the sequence u
k
converges towards an optimal point u
∞ of the linear program. We also study the convergence of the associated dual sequence μ
i
k
=exp[(A
i
u
k
-b
i
)/r
k
] towards a dual optimal solution.
Received: May 2000 / Accepted: November 2001?Published online June 25, 2002 相似文献
2.
The real partE
ℝ of a real Enriques surfaceE admits a natural decomposition in two halves,E
ℝ=E
ℝ
(1)
∪E
ℝ
(2)
, each half being a union of components ofE
ℝ. We classify the triads (E
ℝ;E
ℝ
(1)
,E
ℝ
(2)
) up to homeomorphism. Most results extend to surfaces of more general nature than Enriques surfaces. We use and study in
details the properties of Kalinin's filtration in the homology of the fixed point set of an involution, which is a convenient
tool not widely known in topology of real algebraic varieties. 相似文献
3.
Giuseppina Vannella 《Annali di Matematica Pura ed Applicata》2002,180(4):429-440
We consider a Neumann problem of the type -εΔu+F
′(u(x))=0 in an open bounded subset Ω of R
n
, where F is a real function which has exactly k maximum points.
Using Morse theory we find that, for ε suitably small, there are at least 2k nontrivial solutions of the problem and we give some qualitative information about them.
Received: October 30, 1999 Published online: December 19, 2001 相似文献
4.
We propose a class of non-interior point algorithms for solving the complementarity problems(CP): Find a nonnegative pair
(x,y)∈ℝ
2n
satisfying y=f(x) and x
i
y
i
=0 for every i∈{1,2,...,n}, where f is a continuous mapping from ℝ
n
to ℝ
n
. The algorithms are based on the Chen-Harker-Kanzow-Smale smoothing functions for the CP, and have the following features;
(a) it traces a trajectory in ℝ
3n
which consists of solutions of a family of systems of equations with a parameter, (b) it can be started from an arbitrary
(not necessarily positive) point in ℝ
2n
in contrast to most of interior-point methods, and (c) its global convergence is ensured for a class of problems including
(not strongly) monotone complementarity problems having a feasible interior point. To construct the algorithms, we give a
homotopy and show the existence of a trajectory leading to a solution under a relatively mild condition, and propose a class
of algorithms involving suitable neighborhoods of the trajectory. We also give a sufficient condition on the neighborhoods
for global convergence and two examples satisfying it.
Received April 9, 1997 / Revised version received September 2, 1998? Published online May 28, 1999 相似文献
5.
We investigate certain combinatorial properties of the central curve associated with interior point methods for linear optimization.
We define a measure of complexity for the curve in terms of the number of turns, or changes of direction, that it makes in
a geometric sense, and then perform an average case analysis of this measure for P-matrix linear complementarity problems. We show that the expected number of nondegenerate turns taken by the central curve
is bounded by n
2-n, where the expectation is taken with respect to a sign-invariant probability distribution on the problem data. As an alternative
measure of complexity, we also consider the number of times the central curve intersects with a wide class of algebraic hypersurfaces,
including such objects as spheres and boxes. As an example of the results obtained, we show that the primal and dual variables
in each coordinate of the central curve cross each other at most once, on average. As a further example, we show that the
central curve intersects any sphere centered at the origin at most twice, on average.
Received May 28, 1998 / Revised version received October 12, 1999?Published online December 15, 1999 相似文献
6.
S.M. Natanzon 《Mathematische Zeitschrift》2003,243(2):391-407
Let Y be a complex algebraic curve and let be the set of all real algebraic curves with complexification , such that the real points divide . We find all such families [Y]. According to Harnak theorem a number of connected components of satisfies by the inequality , where g is the genus of Y. We prove that and these estimates are exact.
Received: 15 November 2001; in final form: 28 April 2002/Published online: 2 December 2002 相似文献
7.
8.
Laurent Lafforgue 《Inventiones Mathematicae》2002,147(1):1-241
One proves Langlands’ correspondence for GL
r
over function fields. This is a generalization of Drinfeld’s proof in the case of rank 2 : Langlands’ correspondence is realized
in ℓ-adic cohomology spaces of the modular varieties classifying rank r Drinfeld shtukas.
Oblatum 13-X-2000 & 7-VI-2001?Published online: 12 October 2001 相似文献
9.
U. Luther 《Annali di Matematica Pura ed Applicata》2003,182(2):161-200
We show that the representation theorem for classical approximation spaces can be generalized to spaces A(X,l
q
(ℬ))={f∈X:{E
n
(f)}∈l
q
(ℬ)} in which the weighted l
q
-space l
q
(ℬ) can be (more or less) arbitrary. We use this theorem to show that generalized approximation spaces can be viewed as real
interpolation spaces (defined with K-functionals or main-part K-functionals) between couples of quasi-normed spaces which satisfy certain Jackson and Bernstein-type inequalities. Especially,
interpolation between an approximation space and the underlying quasi-normed space leads again to an approximation space.
Together with a general reiteration theorem, which we also prove in the present paper, we obtain formulas for interpolation
of two generalized approximation spaces.
Received: December 6, 2001; in final form: April 2, 2002?Published online: March 14, 2003 相似文献
10.
A conic linear system is a system of the form?P(d): find x that solves b - Ax∈C
Y
, x∈C
X
,? where C
X
and C
Y
are closed convex cones, and the data for the system is d=(A,b). This system is“well-posed” to the extent that (small) changes in the data (A,b) do not alter the status of the system (the system remains solvable or not). Renegar defined the “distance to ill-posedness”,
ρ(d), to be the smallest change in the data Δd=(ΔA,Δb) for which the system P(d+Δd) is “ill-posed”, i.e., d+Δd is in the intersection of the closure of feasible and infeasible instances d’=(A’,b’) of P(·). Renegar also defined the “condition measure” of the data instance d as C(d):=∥d∥/ρ(d), and showed that this measure is a natural extension of the familiar condition measure associated with systems of linear
equations. This study presents two categories of results related to ρ(d), the distance to ill-posedness, and C(d), the condition measure of d. The first category of results involves the approximation of ρ(d) as the optimal value of certain mathematical programs. We present ten different mathematical programs each of whose optimal
values provides an approximation of ρ(d) to within certain constants, depending on whether P(d) is feasible or not, and where the constants depend on properties of the cones and the norms used. The second category of
results involves the existence of certain inscribed and intersecting balls involving the feasible region of P(d) or the feasible region of its alternative system, in the spirit of the ellipsoid algorithm. These results roughly state that
the feasible region of P(d) (or its alternative system when P(d) is not feasible) will contain a ball of radius r that is itself no more than a distance R from the origin, where the ratio R/r satisfies R/r≤c
1
C(d), and such that r≥ and R≤c
3
C(d), where c
1,c
2,c
3 are constants that depend only on properties of the cones and the norms used. Therefore the condition measure C(d) is a relevant tool in proving the existence of an inscribed ball in the feasible region of P(d) that is not too far from the origin and whose radius is not too small.
Received November 2, 1995 / Revised version received June 26, 1998?Published online May 12, 1999 相似文献
11.
Given a linear transformation L:?
n
→?
n
and a matrix Q∈?
n
, where ?
n
is the space of all symmetric real n×n matrices, we consider the semidefinite linear complementarity problem SDLCP(L,?
n
+,Q) over the cone ?
n
+ of symmetric n×n positive semidefinite matrices. For such problems, we introduce the P-property and its variants, Q- and GUS-properties. For a matrix A∈R
n×n
, we consider the linear transformation L
A
:?
n
→?
n
defined by L
A
(X):=AX+XA
T
and show that the P- and Q-properties for L
A
are equivalent to A being positive stable, i.e., real parts of eigenvalues of A are positive. As a special case of this equivalence, we deduce a theorem of Lyapunov.
Received: March 1999 / Accepted: November 1999?Published online April 20, 2000 相似文献
12.
Krzysztof C. Kiwiel 《Mathematical Programming》2001,90(1):1-25
We study a general subgradient projection method for minimizing a quasiconvex objective subject to a convex set constraint
in a Hilbert space. Our setting is very general: the objective is only upper semicontinuous on its domain, which need not
be open, and various subdifferentials may be used. We extend previous results by proving convergence in objective values and
to the generalized solution set for classical stepsizes t
k
→0, ∑t
k
=∞, and weak or strong convergence of the iterates to a solution for {t
k
}∈ℓ2∖ℓ1 under mild regularity conditions. For bounded constraint sets and suitable stepsizes, the method finds ε-solutions with an
efficiency estimate of O(ε-2), thus being optimal in the sense of Nemirovskii.
Received: October 4, 1998 / Accepted: July 24, 2000?Published online January 17, 2001 相似文献
13.
For a certain class of domains Ω⊂ℂ with smooth boundary and Δtilde;Ω=w
2Δ the Laplace–Beltrami operator with respect to the Poincaré metric ds
2=w(z)-2
dz dz on Ω, we (1) show that the Green function for the biharmonic operator Δtilde;Ω
2, with Dirichlet boundary data, is positive on Ω×Ω; and (2) obtain an eigenfunction expansion for the operator Δtilde;Ω, which reduces to the ordinary non-Euclidean Fourier transform of Helgason for Ω=𝔻 (the unit disc). In both cases the proofs
go via uniformization, and in (1) we obtain a Myrberg-like formula for the corresponding Green function. Finally, the latter
formula as well as the eigenfunction expansion are worked out more explicitly in the simplest case of Ω an annulus, and a
result is established concerning the convergence of the series ∑
ω∈G
(1-|ω0|2)
s
for G the covering group of the uniformization map of Ω and 0<s<1.
Received: August 21, 2000?Published online: October 30, 2002
RID="*"
ID="*"The first author was supported by GA AV CR grants no. A1019701 and A1019005. 相似文献
14.
15.
We classify all isomorphism classes of stable torsionfree sheaves on an irreducible nodal curve of arithmetic genus one defined
over ℂ. Let X be a nodal curve of arithmetic genus one defined over ℝ, with exactly one node, such that X does not have any real points apart from the node. We classify all isomorphism classes of stable real algebraic torsionfree
sheaves over X of even rank. We also classify all isomorphism classes of real algebraic torsionfree sheaves over X of rank one. 相似文献
16.
Our knowledge of linear series on real algebraic curves is still very incomplete. In this paper we restrict to pencils (complete linear series of dimension one). Let X denote a real curve of genus g with real points and let k(R) be the smallest degree of a pencil on X (the real gonality of X). Then we can find on X a base point free pencil of degree g+1 (resp. g if X is not hyperelliptic, i.e. if k(R)>2) with an assigned geometric behaviour w.r.t. the real components of X, and if we prove that which is the same bound as for the gonality of a complex curve of even genus g. Furthermore, if the complexification of X is a k-gonal curve (k≥2) one knows that k≤k(R)≤2k−2, and we show that for any two integers k≥2 and 0≤n≤k−2 there is a real curve with real points and k-gonal complexification such that its real gonality is k+n. 相似文献
17.
François Maucourant 《Israel Journal of Mathematics》2006,152(1):143-155
We prove that almost every (resp. almost no) geodesic rays in a finite volume hyperbolic manifold of real dimensionn intersects for arbitrary large timest a decreasing family of balls of radiusr
t, provided the integral ∫
0
∞
r
t
n
−1 dt diverges (resp. converges). 相似文献
18.
V. A. Krasnov 《Mathematical Notes》1998,64(3):347-350
Real theta characteristics of a real algebraic curve are studied. The numbers of even and odd real theta characteristics are
calculated. These numbers depend on the topological characteristics of the curve only.
Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 403–106, September, 1998. 相似文献
19.
A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities 总被引:17,自引:0,他引:17
In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the
box constrained variational inequalities (BVI). Instead of using an infinite sequence of smoothing approximation functions,
we use a single smoothing approximation function and Robinson’s normal equation to reformulate NCP and BVI as an equivalent
nonsmooth equation H(u,x)=0, where H:ℜ
2n
→ℜ
2n
, u∈ℜ
n
is a parameter variable and x∈ℜ
n
is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {z
k
=(u
k
,x
k
)} such that the mapping H(·) is continuously differentiable at each z
k
and may be non-differentiable at the limiting point of {z
k
}. We prove that three most often used Gabriel-Moré smoothing functions can generate strongly semismooth functions, which
play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods. We do not
require any function value of F or its derivative value outside the feasible region while at each step we only solve a linear system of equations and if
we choose a certain smoothing function only a reduced form needs to be solved. Preliminary numerical results show that the
proposed methods for particularly chosen smoothing functions are very promising.
Received June 23, 1997 / Revised version received July 29, 1999?Published online December 15, 1999 相似文献
20.
We consider the parametric programming problem (Q
p
) of minimizing the quadratic function f(x,p):=x
T
Ax+b
T
x subject to the constraint Cx≤d, where x∈ℝ
n
, A∈ℝ
n×n
, b∈ℝ
n
, C∈ℝ
m×n
, d∈ℝ
m
, and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Q
p
) are denoted by M(p) and M
loc
(p), respectively. It is proved that if the point-to-set mapping M
loc
(·) is lower semicontinuous at p then M
loc
(p) is a nonempty set which consists of at most ?
m,n
points, where ?
m,n
= is the maximal cardinality of the antichains of distinct subsets of {1,2,...,m} which have at most n elements. It is proved also that the lower semicontinuity of M(·) at p implies that M(p) is a singleton. Under some regularity assumption, these necessary conditions become the sufficient ones.
Received: November 5, 1997 / Accepted: September 12, 2000?Published online November 17, 2000 相似文献