共查询到20条相似文献,搜索用时 46 毫秒
1.
Combining search directions using gradient flows 总被引:2,自引:0,他引:2
The efficient combination of directions is a significant problem in line search methods that either use negative curvature,
or wish to include additional information such as the gradient or different approximations to the Newton direction.
In this paper we describe a new procedure to combine several of these directions within an interior-point primal-dual algorithm.
Basically, we combine in an efficient manner a modified Newton direction with the gradient of a merit function and a direction
of negative curvature, if it exists. We also show that the procedure is well-defined, and it has reasonable theoretical properties
regarding the rate of convergence of the method.
We also present numerical results from an implementation of the proposed algorithm on a set of small test problems from the
CUTE collection.
Received: November 2000 / Accepted: October 2002 Published online: February 14, 2003
Key Words. negative curvature – primal-dual methods – interior-point methods – nonconvex optimization – line searches
Mathematics Subject Classification (1991): 49M37, 65K05, 90C30 相似文献
2.
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
3.
Non-monotone trust region methods for nonlinear equality constrained optimization without a penalty function 总被引:3,自引:0,他引:3
We propose and analyze a class of penalty-function-free nonmonotone trust-region methods for nonlinear equality constrained
optimization problems. The algorithmic framework yields global convergence without using a merit function and allows nonmonotonicity
independently for both, the constraint violation and the value of the Lagrangian function. Similar to the Byrd–Omojokun class
of algorithms, each step is composed of a quasi-normal and a tangential step. Both steps are required to satisfy a decrease
condition for their respective trust-region subproblems. The proposed mechanism for accepting steps combines nonmonotone decrease
conditions on the constraint violation and/or the Lagrangian function, which leads to a flexibility and acceptance behavior
comparable to filter-based methods. We establish the global convergence of the method. Furthermore, transition to quadratic
local convergence is proved. Numerical tests are presented that confirm the robustness and efficiency of the approach.
Received: December 14, 2000 / Accepted: August 30, 2001 Published online: September 27, 2002
Key words. nonmonotone trust-region methods – sequential quadratic programming – penalty function – global convergence – equality constraints
– local convergence – large-scale optimization
Mathematics Subject Classification (2000): 65K05, 90C30 相似文献
4.
Stephen J. Wright 《Mathematical Programming》2003,95(1):137-160
In the vicinity of a solution of a nonlinear programming problem at which both strict complementarity and linear independence
of the active constraints may fail to hold, we describe a technique for distinguishing weakly active from strongly active
constraints. We show that this information can be used to modify the sequential quadratic programming algorithm so that it
exhibits superlinear convergence to the solution under assumptions weaker than those made in previous analyses.
Received: December 18, 2000 / Accepted: January 14, 2002 Published online: September 27, 2002
RID="★"
ID="★" Research supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of
Advanced Scientific Computing Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.
Key words. nonlinear programming problems – degeneracy – active constraint identification – sequential quadratic programming 相似文献
5.
Andreas Fischer 《Mathematical Programming》2002,94(1):91-124
An iterative framework for solving generalized equations with nonisolated solutions is presented. For generalized equations
with the structure , where is a multifunction and F is single-valued, the framework covers methods that, at each step, solve subproblems of the type . The multifunction approximates F around s. Besides a condition on the quality of this approximation, two other basic assumptions are employed to show Q-superlinear
or Q-quadratic convergence of the iterates to a solution. A key assumption is the upper Lipschitz-continuity of the solution
set map of the perturbed generalized equation . Moreover, the solvability of the subproblems is required. Conditions that ensure these assumptions are discussed in general
and by means of several applications. They include monotone mixed complementarity problems, Karush-Kuhn-Tucker systems arising
from nonlinear programs, and nonlinear equations. Particular results deal with error bounds and upper Lipschitz-continuity
properties for these problems.
Received: November 2001 / Accepted: November 2002 Published online: December 9, 2002
Key Words. generalized equation – nonisolated solutions – Newton's method – superlinear convergence – upper Lipschitz-continuity – mixed
complementarity problem – error bounds
Mathematics Subject Classification (1991): 90C30, 65K05, 90C31, 90C33 相似文献
6.
A feasible semismooth asymptotically Newton method for mixed complementarity problems 总被引:2,自引:0,他引:2
Semismooth Newton methods constitute a major research area for solving mixed complementarity problems (MCPs). Early research
on semismooth Newton methods is mainly on infeasible methods. However, some MCPs are not well defined outside the feasible
region or the equivalent unconstrained reformulations of other MCPs contain local minimizers outside the feasible region.
As both these problems could make the corresponding infeasible methods fail, more recent attention is on feasible methods.
In this paper we propose a new feasible semismooth method for MCPs, in which the search direction asymptotically converges
to the Newton direction. The new method overcomes the possible non-convergence of the projected semismooth Newton method,
which is widely used in various numerical implementations, by minimizing a one-dimensional quadratic convex problem prior
to doing (curved) line searches.
As with other semismooth Newton methods, the proposed method only solves one linear system of equations at each iteration.
The sparsity of the Jacobian of the reformulated system can be exploited, often reducing the size of the system that must
be solved. The reason for this is that the projection onto the feasible set increases the likelihood of components of iterates
being active. The global and superlinear/quadratic convergence of the proposed method is proved under mild conditions. Numerical
results are reported on all problems from the MCPLIB collection [8].
Received: December 1999 / Accepted: March 2002 Published online: September 5, 2002
RID="★"
ID="★" This work was supported in part by the Australian Research Council.
Key Words. mixed complementarity problems – semismooth equations – projected Newton method – convergence
AMS subject classifications. 90C33, 90C30, 65H10 相似文献
7.
On implementing a primal-dual interior-point method for conic quadratic optimization 总被引:8,自引:0,他引:8
Based on the work of the Nesterov and Todd on self-scaled cones an implementation of a primal-dual interior-point method
for solving large-scale sparse conic quadratic optimization problems is presented. The main features of the implementation
are it is based on a homogeneous and self-dual model, it handles rotated quadratic cones directly, it employs a Mehrotra type
predictor-corrector extension and sparse linear algebra to improve the computational efficiency. Finally, the implementation
exploits fixed variables which naturally occurs in many conic quadratic optimization problems. This is a novel feature for
our implementation. Computational results are also presented to document that the implementation can solve very large problems
robustly and efficiently.
Received: November 18, 2000 / Accepted: January 18, 2001 Published online: September 27, 2002
Key Words. conic optimization – interior-point methods – large-scale implementation 相似文献
8.
In this paper we consider stochastic programming problems where the objective function is given as an expected value of a
convex piecewise linear random function. With an optimal solution of such a problem we associate a condition number which
characterizes well or ill conditioning of the problem. Using theory of Large Deviations we show that the sample size needed
to calculate the optimal solution of such problem with a given probability is approximately proportional to the condition
number.
Received: May 2000 / Accepted: May 2002-07-16 Published online: September 5, 2002
RID="★"
The research of this author was supported, in part, by grant DMS-0073770 from the National Science Foundation
Key Words. stochastic programming – Monte Carlo simulation – large deviations theory – ill-conditioned problems 相似文献
9.
We revise the Volume Algorithm (VA) for linear programming and relate it to bundle methods. When first introduced, VA was
presented as a subgradient-like method for solving the original problem in its dual form. In a way similar to the serious/null
steps philosophy of bundle methods, VA produces green, yellow or red steps. In order to give convergence results, we introduce
in VA a precise measure for the improvement needed to declare a green or serious step. This addition yields a revised formulation
(RVA) that is halfway between VA and a specific bundle method, that we call BVA. We analyze the convergence properties of
both RVA and BVA. Finally, we compare the performance of the modified algorithms versus VA on a set of Rectilinear Steiner
problems of various sizes and increasing complexity, derived from real world VLSI design instances.
Received: December 1999 / Accepted: September 2002 Published online: December 19, 2002
Key Words. volume algorithm – bundle methods – Steiner problems
Correspondence to: Claudia A. Sagastizábal, e-mail: sagastiz@impa.br 相似文献
10.
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
11.
Recently, interior-point algorithms have been applied to nonlinear and nonconvex optimization. Most of these algorithms are
either primal-dual path-following or affine-scaling in nature, and some of them are conjectured to converge to a local minimum.
We give several examples to show that this may be untrue and we suggest some strategies for overcoming this difficulty.
Received: June 26, 2000 / Accepted: April 2002 Published online: September 5, 2002
Key words. Nonconvex quadratic optimization – local minimum – interior-point algorithms – trust region – branch-and-cut
This research is supported by the National Science Foundation Grant CCR-9731273 and DMS-9703490. 相似文献
12.
In this paper, we describe how to reformulate a problem that has second-order cone and/or semidefiniteness constraints in
order to solve it using a general-purpose interior-point algorithm for nonlinear programming. The resulting problems are smooth
and convex, and numerical results from the DIMACS Implementation Challenge problems and SDPLib are provided.
Received: March 10, 2001 / Accepted: January 18, 2002 Published online: September 27, 2002
Key Words. semidefinite programming – second-order cone programming – interior-point methods – nonlinear programming
Mathematics Subject Classification (2000): 20E28, 20G40, 20C20 相似文献
13.
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
14.
Renato. D. C. Monteiro 《Mathematical Programming》2003,97(1-2):209-244
In this paper, we survey the most recent methods that have been developed for the solution of semidefinite programs. We first
concentrate on the methods that have been primarily motivated by the interior point (IP) algorithms for linear programming,
putting special emphasis in the class of primal-dual path-following algorithms. We also survey methods that have been developed
for solving large-scale SDP problems. These include first-order nonlinear programming (NLP) methods and more specialized path-following
IP methods which use the (preconditioned) conjugate gradient or residual scheme to compute the Newton direction and the notion
of matrix completion to exploit data sparsity.
Received: December 16, 2002 / Accepted: May 5, 2003
Published online: May 28, 2003
Key words. semidefinite programming – interior-point methods – polynomial complexity – path-following methods – primal-dual methods
– nonlinear programming – Newton method – first-order methods – bundle method – matrix completion
The author's research presented in this survey article has been supported in part by NSF through grants INT-9600343, INT-9910084,
CCR-9700448, CCR-9902010, CCR-0203113 and ONR through grants N00014-93-1-0234, N00014-94-1-0340 and N00014-03-1-0401.
Mathematics Subject Classification (2000): 65K05, 90C06, 90C22, 90C25, 90C30, 90C51 相似文献
15.
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
16.
Solving semidefinite-quadratic-linear programs using SDPT3 总被引:3,自引:1,他引:2
This paper discusses computational experiments with linear optimization problems involving semidefinite, quadratic, and linear
cone constraints (SQLPs). Many test problems of this type are solved using a new release of SDPT3, a Matlab implementation
of infeasible primal-dual path-following algorithms. The software developed by the authors uses Mehrotra-type predictor-corrector
variants of interior-point methods and two types of search directions: the HKM and NT directions. A discussion of implementation
details is provided and computational results on problems from the SDPLIB and DIMACS Challenge collections are reported.
Received: March 19, 2001 / Accepted: January 18, 2002 Published online: October 9, 2002
Mathematics Subject Classification (2000): 90C05, 90C22 相似文献
17.
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
18.
The stability number α(G) for a given graph G is the size of a maximum stable set in G. The Lovász theta number provides an upper bound on α(G) and can be computed in polynomial time as the optimal value of the Lovász semidefinite program. In this paper, we show that
restricting the matrix variable in the Lovász semidefinite program to be rank-one and rank-two, respectively, yields a pair
of continuous, nonlinear optimization problems each having the global optimal value α(G). We propose heuristics for obtaining large stable sets in G based on these new formulations and present computational results indicating the effectiveness of the heuristics.
Received: December 13, 2000 / Accepted: September 3, 2002 Published online: December 19, 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.
Key Words. maximum stable set – maximum clique – minimum vertex cover – semidefinite program – semidefinite relaxation – continuous
optimization heuristics – nonlinear programming
Mathematics Subject Classification (2000): 90C06, 90C27, 90C30 相似文献
19.
Wensheng Wang 《Probability Theory and Related Fields》2003,126(2):203-220
Kesten and Spitzer have shown that certain random walks in random sceneries converge to stable processes in random sceneries.
In this paper, we consider certain random walks in sceneries defined using stationary Gaussian sequence, and show their convergence
towards a certain self-similar process that we call fractional Brownian motion in Brownian scenery.
Received: 17 April 2002 / Revised version: 11 October 2002 /
Published online: 15 April 2003
Research supported by NSFC (10131040).
Mathematics Subject Classification (2002): 60J55, 60J15, 60J65
Key words or phrases: Weak convergence – Random walk in random scenery – Local time – Fractional Brownian motion in Brownian scenery 相似文献
20.
We introduce a new upper bound for the maximum-entropy sampling problem. Our bound is described as the solution of a linear
integer program. The bound depends on a partition of the underlying set of random variables. For the case in which each block
of the partition has bounded cardinality, we describe an efficient dynamic-programming algorithm to calculate the bound. For
the very special case in which the blocks have no more than two elements, we describe an efficient algorithm for calculating
the bound based on maximum-weight matching. This latter formulation has particular value for local-search procedures that
seek to find a good partition. We relate our bound to recent bounds of Hoffman, Lee and Williams. Finally, we report on the
results of some computational experiments.
Received: September 27, 2000 / Accepted: July 26, 2001 Published online: September 5, 2002
Key words. experimental design – design of experiments – entropy – maximum-entropy sampling – matching – integer program – spectral
bound – Fischer's inequality – branch-and-bound – dynamic programming
Mathematics Subject Classification (2000): 52B12, 90C10
Send offprint requests to: Jon Lee
Correspondence to: Jon Lee 相似文献