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1.
Leonardo Colzani Alice Cominardi Krzysztof Stempak 《Annali di Matematica Pura ed Applicata》2002,181(1):25-54
We study some boundedness properties of radial solutions to the Cauchy problem associated to the wave equation (∂
t
2-▵
x
)u(t,x)=0 and meanwhile we give a new proof of the solution formula.
Received: July 7, 1998?Published online: March 19, 2002 相似文献
2.
Entropic proximal decomposition methods for convex programs and variational inequalities 总被引:2,自引:0,他引:2
We consider convex optimization and variational inequality problems with a given separable structure. We propose a new decomposition
method for these problems which combines the recent logarithmic-quadratic proximal theory introduced by the authors with a
decomposition method given by Chen-Teboulle for convex problems with particular structure. The resulting method allows to
produce for the first time provably convergent decomposition schemes based on C
∞ Lagrangians for solving convex structured problems. Under the only assumption that the primal-dual problems have nonempty
solution sets, global convergence of the primal-dual sequences produced by the algorithm is established.
Received: October 6, 1999 / Accepted: February 2001?Published online September 17, 2001 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.
Inexact implicit methods for monotone general variational inequalities 总被引:32,自引:0,他引:32
Bingsheng He 《Mathematical Programming》1999,86(1):199-217
Solving a variational inequality problem is equivalent to finding a solution of a system of nonsmooth equations. Recently,
we proposed an implicit method, which solves monotone variational inequality problem via solving a series of systems of nonlinear
smooth (whenever the operator is smooth) equations. It can exploit the facilities of the classical Newton–like methods for
smooth equations. In this paper, we extend the method to solve a class of general variational inequality problems Moreover, we improve the implicit method to allow inexact solutions of the systems of nonlinear equations at each iteration.
The method is shown to preserve the same convergence properties as the original implicit method.
Received July 31, 1995 / Revised version received January 15, 1999? Published online May 28, 1999 相似文献
5.
In this paper we study the existence of global solutions to the Euler equations of compressible isothermal gas dynamics with
semiconductor devices. We construct the approximate solutions by Lax–Friedrichs scheme. The convergence and consistency are
obtained by using the compensated compactness framework for γ = 1. The global entropy solutions in L∞ are obtained. We deal with the initial data containing unbounded velocity which is different from the isentropic case.
Received: November 18, 2003 相似文献
6.
Based on the authors’ previous work which established theoretical foundations of two, conceptual, successive convex relaxation
methods, i.e., the SSDP (Successive Semidefinite Programming) Relaxation Method and the SSILP (Successive Semi-Infinite Linear Programming)
Relaxation Method, this paper proposes their implementable variants for general quadratic optimization problems. These problems
have a linear objective function c
T
x to be maximized over a nonconvex compact feasible region F described by a finite number of quadratic inequalities. We introduce two new techniques, “discretization” and “localization,”
into the SSDP and SSILP Relaxation Methods. The discretization technique makes it possible to approximate an infinite number
of semi-infinite SDPs (or semi-infinite LPs) which appeared at each iteration of the original methods by a finite number of
standard SDPs (or standard LPs) with a finite number of linear inequality constraints. We establish:?•Given any open convex set U containing F, there is an implementable discretization of the SSDP (or SSILP) Relaxation Method
which generates a compact convex set C such that F⊆C⊆U in a finite number of iterations.?The localization technique is for the cases where we are only interested in upper bounds on the optimal objective value (for
a fixed objective function vector c) but not in a global approximation of the convex hull of F. This technique allows us to generate a convex relaxation of F that is accurate only in certain directions in a neighborhood of the objective direction c. This cuts off redundant work to make the convex relaxation accurate in unnecessary directions. We establish:?•Given any positive number ε, there is an implementable localization-discretization of the SSDP (or SSILP) Relaxation Method
which generates an upper bound of the objective value within ε of its maximum in a finite number of iterations.
Received: June 30, 1998 / Accepted: May 18, 2000?Published online September 20, 2000 相似文献
7.
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
8.
We modify and extend proofs of Serrin’s symmetry result for overdetermined boundary value problems from the Laplace-operator
to a general quasilinear operator and remove a strong ellipticity assumption in Philippin (Maximum principles and eigenvalue
problems in partial differential equations (Knoxville, TN, 1987), Longman Sci. Tech., Pitman Res. Notes Math. Ser., Harlow,
175, pp. 34–48, 1988) and a growth assumption in Garofalo and Lewis (A symmetry result related to some overdetermined boundary
value problems, Am. J. Math. 111, 9–33, 1989) on the diffusion coefficient A, as well as a starshapedness assumption on Ω in Fragalà et al. (Overdetermined boundary value problems with possibly degenerate
ellipticity: a geometric approach. Math. Zeitschr. 254, 117–132, 2006). 相似文献
9.
A class of affine-scaling interior-point methods for bound constrained optimization problems is introduced which are locally
q–superlinear or q–quadratic convergent. It is assumed that the strong second order sufficient optimality conditions at the
solution are satisfied, but strict complementarity is not required. The methods are modifications of the affine-scaling interior-point
Newton methods introduced by T. F. Coleman and Y. Li (Math. Programming, 67, 189–224, 1994). There are two modifications. One is a modification of the scaling matrix, the other one is the use of a
projection of the step to maintain strict feasibility rather than a simple scaling of the step. A comprehensive local convergence
analysis is given. A simple example is presented to illustrate the pitfalls of the original approach by Coleman and Li in
the degenerate case and to demonstrate the performance of the fast converging modifications developed in this paper.
Received October 2, 1998 / Revised version received April 7, 1999?Published online July 19, 1999 相似文献
10.
A.B. Levy 《Mathematical Programming》1999,85(2):397-406
n such that x≥0, F(x,u)-v≥0 , and F(x,u)-v T·x=0 where these are vector inequalities. We characterize the local upper Lipschitz continuity of the (possibly set-valued)
solution mapping which assigns solutions x to each parameter pair (v,u). We also characterize when this solution mapping is
locally a single-valued Lipschitzian mapping (so solutions exist, are unique, and depend Lipschitz continuously on the parameters).
These characterizations are automatically sufficient conditions for the more general (and usual) case where v=0. Finally,
we study the differentiability properties of the solution mapping in both the single-valued and set-valued cases, in particular
obtaining a new characterization of B-differentiability in the single-valued case, along with a formula for the B-derivative.
Though these results cover a broad range of stability properties, they are all derived from similar fundamental principles
of variational analysis.
Received March 30, 1998 / Revised version received July 21, 1998
Published online January 20, 1999 相似文献
11.
Paulo Amorim Wladimir Neves 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(15):4898-4917
We consider a hyperbolic conservation law posed on an (N+1)-dimensional spacetime, whose flux is a field of differential forms of degree N. Generalizing the classical Kuznetsov’s method, we derive an L1 error estimate which applies to a large class of approximate solutions. In particular, we apply our main theorem and deal with two entropy solutions associated with distinct flux fields, as well as with an entropy solution and an approximate solution. Our framework encompasses, for instance, equations posed on a globally hyperbolic Lorentzian manifold. 相似文献
12.
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
13.
Futoshi Takahashi 《Calculus of Variations and Partial Differential Equations》2007,29(4):509-520
We continue to study the asymptotic behavior of least energy solutions to the following fourth order elliptic problem (E
p
): as p gets large, where Ω is a smooth bounded domain in R
4
. In our earlier paper (Takahashi in Osaka J. Math., 2006), we have shown that the least energy solutions remain bounded uniformly
in p and they have one or two “peaks” away form the boundary. In this note, following the arguments in Adimurthi and Grossi (Proc.
AMS 132(4):1013–1019, 2003) and Lin and Wei (Comm. Pure Appl. Math. 56:784–809, 2003), we will obtain more sharper estimates
of the upper bound of the least energy solutions and prove that the least energy solutions must develop single-point spiky
pattern, under the assumption that the domain is convex. 相似文献
14.
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
15.
16.
Diethard Klatte 《Mathematical Programming》2000,88(2):285-311
We analyze the local upper Lipschitz behavior of critical points, stationary solutions and local minimizers to parametric
C
1,1 programs. In particular, we derive a characterization of this property for the stationary solution set map without assuming
the Mangasarian–Fromovitz CQ. Moreover, conditions which also ensure the persistence of solvability are given, and the special
case of linear constraints is handled. The present paper takes pattern from [21] by continuing the approach via contingent
derivatives of the Kojima function associated with the given optimization problem.
Received: June 10, 1999 / Accepted: November 15, 1999?Published online July 20, 2000 相似文献
17.
João Paulo Dias 《Journal of Differential Equations》2011,251(3):492-503
We consider a system coupling a multidimensional semilinear Schrödinger equation and a multidimensional nonlinear scalar conservation law with viscosity, which is motivated by a model of short wave-long wave interaction introduced by Benney (1977). We prove the global existence and uniqueness of the solution of the Cauchy problem for this system. We also prove the convergence of the whole sequence of solutions when the viscosity ε and the interaction parameter α approach zero so that α=o(ε1/2). We also indicate how to extend these results to more general systems which couple multidimensional semilinear systems of Schrödinger equations with multidimensional nonlinear systems of scalar conservation laws mildly coupled. 相似文献
18.
We consider optimality systems of Karush-Kuhn-Tucker (KKT) type, which arise, for example, as primal-dual conditions characterizing
solutions of optimization problems or variational inequalities. In particular, we discuss error bounds and Newton-type methods
for such systems. An exhaustive comparison of various regularity conditions which arise in this context is given. We obtain
a new error bound under an assumption which we show to be strictly weaker than assumptions previously used for KKT systems,
such as quasi-regularity or semistability (equivalently, the R
0-property). Error bounds are useful, among other things, for identifying active constraints and developing efficient local
algorithms. We propose a family of local Newton-type algorithms. This family contains some known active-set Newton methods,
as well as some new methods. Regularity conditions required for local superlinear convergence compare favorably with convergence
conditions of nonsmooth Newton methods and sequential quadratic programming methods.
Received: December 10, 2001 / Accepted: July 28, 2002 Published online: February 14, 2003
Key words. KKT system – regularity – error bound – active constraints – Newton method
Mathematics Subject Classification (1991): 90C30, 65K05 相似文献
19.
Basis- and partition identification for quadratic programming and linear complementarity problems 总被引:1,自引:0,他引:1
Arjan B. Berkelaar Benjamin Jansen Kees Roos Tamás Terlaky 《Mathematical Programming》1999,86(2):261-282
Optimal solutions of interior point algorithms for linear and quadratic programming and linear complementarity problems provide
maximally complementary solutions. Maximally complementary solutions can be characterized by optimal partitions. On the other
hand, the solutions provided by simplex–based pivot algorithms are given in terms of complementary bases. A basis identification
algorithm is an algorithm which generates a complementary basis, starting from any complementary solution. A partition identification
algorithm is an algorithm which generates a maximally complementary solution (and its corresponding partition), starting from
any complementary solution. In linear programming such algorithms were respectively proposed by Megiddo in 1991 and Balinski
and Tucker in 1969. In this paper we will present identification algorithms for quadratic programming and linear complementarity
problems with sufficient matrices. The presented algorithms are based on the principal pivot transform and the orthogonality
property of basis tableaus.
Received April 9, 1996 / Revised version received April 27, 1998?
Published online May 12, 1999 相似文献
20.
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