共查询到10条相似文献,搜索用时 125 毫秒
1.
Non-Interior continuation methods for solving semidefinite complementarity problems 总被引:13,自引:0,他引:13
There recently has been much interest in non-interior continuation/smoothing methods for solving linear/nonlinear complementarity
problems. We describe extensions of such methods to complementarity problems defined over the cone of block-diagonal symmetric
positive semidefinite real matrices. These extensions involve the Chen-Mangasarian class of smoothing functions and the smoothed
Fischer-Burmeister function. Issues such as existence of Newton directions, boundedness of iterates, global convergence, and
local superlinear convergence will be studied. Preliminary numerical experience on semidefinite linear programs is also reported.
Received: October 1999 / Accepted: April 2002 Published online: December 19, 2002
RID="⋆"
ID="⋆" This research is supported by National Science Foundation Grant CCR-9731273.
Key words. semidefinite complementarity problem – smoothing function – non-interior continuation – global convergence – local superlinear
convergence 相似文献
2.
The authors of this paper recently introduced a transformation [4] that converts a class of semidefinite programs (SDPs)
into nonlinear optimization problems free of matrix-valued constraints and variables. This transformation enables the application
of nonlinear optimization techniques to the solution of certain SDPs that are too large for conventional interior-point methods
to handle efficiently. Based on the transformation, we proposed a globally convergent, first-order (i.e., gradient-based)
log-barrier algorithm for solving a class of linear SDPs. In this paper, we discuss an efficient implementation of the proposed
algorithm and report computational results on semidefinite relaxations of three types of combinatorial optimization problems.
Our results demonstrate that the proposed algorithm is indeed capable of solving large-scale SDPs and is particularly effective
for problems with a large number of constraints.
Received: June 22, 2001 / Accepted: January 20, 2002 Published online: December 9, 2002
RID="†"
ID="†"Computational results reported in this paper were obtained on an SGI Origin2000 computer at Rice University acquired
in part with support from NSF Grant DMS-9872009.
RID="⋆"
ID="⋆"This author was supported in part by NSF Grants CCR-9902010, INT-9910084 and CCR-0203426
RID="⋆⋆"
ID="⋆⋆"This author was supported in part by NSF Grants CCR-9902010, INT-9910084 and CCR-0203113
RID="⋆⋆⋆"
ID="⋆⋆⋆"This author was supported in part by DOE Grant DE-FG03-97ER25331, DOE/LANL Contract 03891-99-23 and NSF Grant DMS-9973339.
Key Words. semidefinite program – semidefinite relaxation – nonlinear programming – interior-point methods – limited memory quasi-Newton
methods.
Mathematics Subject Classification (1991): 90C06, 90C27, 90C30. 相似文献
3.
Error bounds for proximal point subproblems and associated inexact proximal point algorithms 总被引:1,自引:0,他引:1
We study various error measures for approximate solution of proximal point regularizations of the variational inequality problem,
and of the closely related problem of finding a zero of a maximal monotone operator. A new merit function is proposed for
proximal point subproblems associated with the latter. This merit function is based on Burachik-Iusem-Svaiter’s concept of
ε-enlargement of a maximal monotone operator. For variational inequalities, we establish a precise relationship between the
regularized gap function, which is a natural error measure in this context, and our new merit function. Some error bounds
are derived using both merit functions for the corresponding formulations of the proximal subproblem. We further use the regularized
gap function to devise a new inexact proximal point algorithm for solving monotone variational inequalities. This inexact
proximal point method preserves all the desirable global and local convergence properties of the classical exact/inexact method,
while providing a constructive error tolerance criterion, suitable for further practical applications. The use of other tolerance
rules is also discussed.
Received: April 28, 1999 / Accepted: March 24, 2000?Published online July 20, 2000 相似文献
4.
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
5.
Stephen M. Robinson 《Mathematical Programming》2003,97(1-2):245-265
This is an expository paper about the analysis of variational conditions over sets defined in finite-dimensional spaces by
fairly smooth functions satisfying a constraint qualification. The primary focus is on results that can provide quantitative
and computable sensitivity information for particular instances of the problems under study, and our objective is to give
a personal view of the state of current knowledge in this area and of gaps in that knowledge that require future work. The
writing style is informal, in keeping with the objective of focusing the reader's attention on the basic concepts and the
relationships between them, rather than on details of the particular results themselves.
Received: December 1, 2002 / Accepted: April 25, 2003
Published online: May 28, 2003
Key words. variational condition – variational inequality – complementarity – sensitivity – stability – nondegeneracy
Mathematics Subject Classification (2000): Primary: 90C31. Secondary: 47J20, 49J40, 49J53, 90C33 相似文献
6.
Index information algorithm with local tuning for solving multidimensional global optimization problems with multiextremal constraints 总被引:2,自引:0,他引:2
Multidimensional optimization problems where the objective function and the constraints are multiextremal non-differentiable
Lipschitz functions (with unknown Lipschitz constants) and the feasible region is a finite collection of robust nonconvex
subregions are considered. Both the objective function and the constraints may be partially defined. To solve such problems
an algorithm is proposed, that uses Peano space-filling curves and the index scheme to reduce the original problem to a H?lder
one-dimensional one. Local tuning on the behaviour of the objective function and constraints is used during the work of the
global optimization procedure in order to accelerate the search. The method neither uses penalty coefficients nor additional
variables. Convergence conditions are established. Numerical experiments confirm the good performance of the technique.
Received: April 2002 / Accepted: December 2002
Published online: March 21, 2003
RID="⋆"
ID="⋆" This research was supported by the following grants: FIRB RBNE01WBBB, FIRB RBAU01JYPN, and RFBR 01–01–00587.
Key Words. global optimization – multiextremal constraints – local tuning – index approach 相似文献
7.
In this paper we consider the problem
where B is a ball in R
n
. For a small d>0, we show the uniqueness (up to rotation) of the one-bubbling solution which concentrates at a point of the boundary.
Received: 12 December 2001 / Published online: 10 February 2003
RID="⋆"
ID="⋆" Supported by M.U.R.S.T., project: ``Variational methods and nonlinear differential equations'
RID="⋆⋆"
ID="⋆⋆" Partial supported by National Center for Theoretical Sciences of NSC, Taiwan
Mathematics Subject Classification (2000): 35J60 相似文献
8.
We show that knowing the displacement-to-traction map associated to the equations of isotropic elastodynamics with residual
stress we can determine the lens maps of compressional and shear waves. We derive several consequences of this for the inverse
problem of determining the residual stress and the Lamé parameters from the displacement-to-traction map.
Received: 6 December 2001 / Revised version: 29 October 2002 /
Published online: 8 April 2003
RID="⋆"
ID="⋆" The author thanks the Department of Mathematics at the University of Washington for its hospitality during his visit
in fall 2000.
RID="⋆⋆"
ID="⋆⋆" Partly supported by NSF grant DMS-0070488 and a John Simon Guggenheim fellowship. The author also thanks MSRI for
partial support and for providing a very stimulating environment during the inverse problems program in fall 2001. 相似文献
9.
Tadahisa Funaki 《Probability Theory and Related Fields》2003,126(2):155-183
We consider random evolution of an interface on a hard wall under periodic boundary conditions. The dynamics are governed
by a system of stochastic differential equations of Skorohod type, which is Langevin equation associated with massless Hamiltonian
added a strong repelling force for the interface to stay over the wall. We study its macroscopic behavior under a suitable
large scale space-time limit and derive a nonlinear partial differential equation, which describes the mean curvature motion
except for some anisotropy effects, with reflection at the wall. Such equation is characterized by an evolutionary variational
inequality.
Received: 10 January 2002 / Revised version: 18 August 2002 /
Published online: 15 April 2003
Mathematics Subject Classification (2000): 60K35, 82C24, 35K55, 35K85
Key words or phrases: Hydrodynamic limit – Effective interfaces – Hard wall – Skorohod's stochastic differential equation – Evolutionary variational
inequality 相似文献
10.
In this paper a new class of proximal-like algorithms for solving monotone inclusions of the form T(x)∋0 is derived. It is obtained by applying linear multi-step methods (LMM) of numerical integration in order to solve the
differential inclusion , which can be viewed as a generalization of the steepest decent method for a convex function. It is proved that under suitable
conditions on the parameters of the LMM, the generated sequence converges weakly to a point in the solution set T
−1
(0). The LMM is very similar to the classical proximal point algorithm in that both are based on approximately evaluating
the resolvants of T. Consequently, LMM can be used to derive multi-step versions of many of the optimization methods based on the classical proximal
point algorithm. The convergence analysis allows errors in the computation of the iterates, and two different error criteria
are analyzed, namely, the classical scheme with summable errors, and a recently proposed more constructive criterion.
Received: April 2001 / Accepted: November 2002
Published online: February 14, 2003
Key Words. proximal point algorithm – monotone operator – numerical integration – strong stability – relative error criterion
Mathematics Subject Classification (1991): 20E28, 20G40, 20C20 相似文献