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1.
Semidefinite programming relaxations for semialgebraic problems 总被引:15,自引:0,他引:15
Pablo A. Parrilo 《Mathematical Programming》2003,96(2):293-320
A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of
polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite
programming conditions that prove infeasibility. The main tools employed are a semidefinite programming formulation of the
sum of squares decomposition for multivariate polynomials, and some results from real algebraic geometry. The techniques provide
a constructive approach for finding bounded degree solutions to the Positivstellensatz, and are illustrated with examples
from diverse application fields.
Received: May 10, 2001 / Accepted May 2002
Published online: April 10, 2003
Key Words. semidefinite programming – convex optimization – sums of squares – polynomial equations – real algebraic geometry
The majority of this research has been carried out while the author was with the Control & Dynamical Systems Department,
California Institute of Technology, Pasadena, CA 91125, USA. 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.
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
4.
Kazuhide Nakata Katsuki Fujisawa Mituhiro Fukuda Masakazu Kojima Kazuo Murota 《Mathematical Programming》2003,95(2):303-327
In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over
all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primal-dual interior-point methods.
This framework is based on some results about positive semidefinite matrix completion, and it can be embodied in two different
ways. One is by a conversion of a given sparse SDP having a large scale positive semidefinite matrix variable into an SDP
having multiple but smaller positive semidefinite matrix variables. The other is by incorporating a positive definite matrix
completion itself in a primal-dual interior-point method. The current article presents the details of their implementations.
We introduce new techniques to deal with the sparsity through a clique tree in the former method and through new computational
formulae in the latter one. Numerical results over different classes of SDPs show that these methods can be very efficient
for some problems.
Received: March 18, 2001 / Accepted: May 31, 2001 Published online: October 9, 2002
RID="⋆"
ID="⋆"The author was supported by The Ministry of Education, Culture, Sports, Science and Technology of Japan.
Key Words. semidefinite programming – primal-dual interior-point method – matrix completion problem – clique tree – numerical results
Mathematics Subject Classification (2000): 90C22, 90C51, 05C50, 05C05 相似文献
5.
The chain rule – fundamental to any kind of analytical differentiation - can be applied in various ways to computational
graphs representing vector functions. These variants result in different operations counts for the calculation of the corresponding
Jacobian matrices. The minimization of the number of arithmetic operations required for the calculation of the complete Jacobian
leads to a hard combinatorial optimization problem.
We will describe an approach to the solution of this problem that builds on the idea of optimizing chained matrix products
using dynamic programming techniques. Reductions by a factor of 3 and more are possible regarding the operations count for
the Jacobian accumulation.
After discussing the mathematical basics of Automatic Differentiation we will show how to compute Jacobians by chained sparse
matrix products. These matrix chains can be reordered, must be pruned, and are finally subject to a dynamic programming algorithm
to reduce the number of scalar multiplications performed.
Received: January 17, 2002 / Accepted: May 29, 2002 Published online: February 14, 2003
Key words. chained matrix product – combinatorial optimization – dynamic programming – edge elimination in computational graphs 相似文献
6.
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
7.
In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common nonsmooth functions arising in the applications such as the Euclidean norm, the maximum eigenvalue function and the least squares functions with ? 1-regularization or elastic net regularization used in statistics and compressed sensing. We show that, under commonly used strict feasibility conditions, the optimal value and an optimal solution of SOS-convex semialgebraic programs can be found by solving a single semidefinite programming problem (SDP). We achieve the results by using tools from semialgebraic geometry, convex-concave minimax theorem and a recently established Jensen inequality type result for SOS-convex polynomials. As an application, we show that robust SOS-convex optimization proble ms under restricted spectrahedron data uncertainty enjoy exact SDP relaxations. This extends the existing exact SDP relaxation result for restricted ellipsoidal data uncertainty and answers an open question in the literature on how to recover a robust solution of uncertain SOS-convex polynomial programs from its semidefinite programming relaxation in this broader setting. 相似文献
8.
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
9.
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
10.
H.D. Mittelmann 《Mathematical Programming》2003,95(2):407-430
This work reports the results of evaluating all computer codes submitted to the Seventh DIMACS Implementation Challenge on
Semidefinite and Related Optimization Problems. The codes were run on a standard platform and on all the benchmark problems
provided by the organizers of the challenge. A total of ten codes were tested on fifty problems in twelve categories. For
each code the most important information is summarized. Together with the tabulated and commented benchmarking results this
provides an overview of the state of the art in this field.
Received: March 27, 2001 / Accepted: April 5, 2002 Published online: October 9, 2002
Key Words. semidefinite programming – second order cone programming – optimization software – performance evaluation
This work is supported in part by grant NSF-CISE-9981984 相似文献
11.
A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization 总被引:7,自引:0,他引:7
In this paper, we present a nonlinear programming algorithm for solving semidefinite programs (SDPs) in standard form. The
algorithm's distinguishing feature is a change of variables that replaces the symmetric, positive semidefinite variable X of the SDP with a rectangular variable R according to the factorization X=RR
T
. The rank of the factorization, i.e., the number of columns of R, is chosen minimally so as to enhance computational speed while maintaining equivalence with the SDP. Fundamental results
concerning the convergence of the algorithm are derived, and encouraging computational results on some large-scale test problems
are also presented.
Received: March 22, 2001 / Accepted: August 30, 2002 Published online: December 9, 2002
Key Words. semidefinite programming – low-rank factorization – nonlinear programming – augmented Lagrangian – limited memory BFGS
This research was supported in part by the National Science Foundation under grants CCR-9902010, INT-9910084, CCR-0203426
and CCR-0203113 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.
Recent advances in the solution of quadratic assignment problems 总被引:6,自引:0,他引:6
Kurt M. Anstreicher 《Mathematical Programming》2003,97(1-2):27-42
The quadratic assignment problem (QAP) is notoriously difficult for exact solution methods. In the past few years a number
of long-open QAPs, including those posed by Steinberg (1961), Nugent et al. (1968) and Krarup (1972) were solved to optimality
for the first time. The solution of these problems has utilized both new algorithms and novel computing structures. We describe
these developments, as well as recent work which is likely to result in the solution of even more difficult instances.
Received: February 13, 2003 / Accepted: April 22, 2003
Published online: May 28, 2003
Key Words. quadratic assignment problem – discrete optimization – branch and bound
Mathematics Subject Classification (1991): 90C27, 90C09, 90C20 相似文献
14.
C. Helmberg 《Mathematical Programming》2003,95(2):381-406
We report numerical results for SBmethod – a publically available implementation of the spectral bundle method – applied
to the 7$^{th}$ DIMACS challenge test sets that are semidefinite relaxations of combinatorial optimization problems. The performance
of the code is heavily influenced by parameters that control bundle update and eigenvalue computation. Unfortunately, no mathematically
sound guidelines for setting them are known. Based on our experience with SBmethod, we propose heuristics for dynamically
updating the parameters as well as a heuristc for improving the starting point. These are now the default settings of SBmethod
Version 1.1. We compare their performance on the DIMACS instances to our previous best choices for Version 1.0. SBmethod Version
1.1 is also part of the independent DIMACS benchmark by H. Mittelmann. Based on these results we try to analyze strengths
and weaknesses of our approach in comparison to other codes for large scale semidefinite programming.
Received: June 21, 2001 / Accepted: April 19, 2002 Published online: September 27, 2002
Mathematics Subject Classification (2000): 90C22, 90C06 相似文献
15.
Graph partition is used in the telecommunication industry to subdivide a transmission network into small clusters. We consider
both linear and semidefinite relaxations for the equipartition problem and present numerical results on real data from France
Telecom networks with up 900 nodes, and also on randomly generated problems.
Received: August 8, 2001 / Accepted: November 9, 2001 Published online: December 9, 2002
RID="★★"
ID="★★" This research was carried out while this author was working at France Telecom R & D, 38–40 rue du Général Leclerc,
F-92794 Issy-Les-Moulineaux Cedex 9, France.
RID="★"
ID="★" This author greatfully acknowledges financial support from the Austrian Science Foundation FWF Project P12660-MAT.
Key words. graph partitioning – semidefinite programming 相似文献
16.
Via the Linking Theorem and Pseudo-index theory, we consider the existence and multiplicity of nontrivial solutions for a
class of elliptic problems in all of ℝ
N
with indefinite linear part involving resonance and non-resonance at any eigenvalue.
Received: 9 September 2002 / Revised version: 14 February 2003
Published online: 24 April 2003
Mathematics Subject Classification (2000): 35J20, 35J70 相似文献
17.
Weakly hyperbolic involutions are introduced and a proof is given of the following local–global principle: a central simple
algebra with involution of any kind is weakly hyperbolic if and only if its signature is zero for all orderings of the ground
field. Also, the order of a weakly hyperbolic algebra with involution is a power of two, this being a direct consequence of
a result of Scharlau. As a corollary an analogue of Pfister's local–global principle is obtained for the Witt group of hermitian
forms over an algebra with involution.
Received: 29 October 2001; in final form: 9 August 2002 /
Published online: 16 May 2003
Mathematics Subject Classification (2000): 16K20, 11E39 相似文献
18.
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
19.
We define the contact boundary of a complex polynomial f : ℂ
n
→ ℂ as the intersection of some generic fiber with a large sphere. We show that, up to contact isotopy, this does not depend
on the choice of the fiber (provided it is generic) and is invariant under polynomial automorphism of ℂ
n
. We next prove that the formal homotopy class of this contact boundary is invariant in a large family of deformations of
polynomials, which are not necessarily topologically trivial.
Received: 15 November 2002
Published online: 20 March 2003
Mathematics Subject Classification (2000): 32S55, 53D15, 32S50 相似文献
20.
Mikael Rönnqvist 《Mathematical Programming》2003,97(1-2):267-284
Optimization models and methods have been used extensively in the forest industry. In this paper we describe the general
wood-flow in forestry and a variety of planning problems. These cover planning periods from a fraction of a second to more
than one hundred years. The problems are modelled using linear, integer and nonlinear models. Solution methods used depend
on the required solution time and include for example dynamic programming, LP methods, branch & bound methods, heuristics
and column generation. The importance of modelling and qualitative information is also discussed.
Received: January 15, 2003 / Accepted: April 24, 2003
Published online: May 28, 2003
Key words. Forestry – modelling – routing – transportation – production planning
Mathematics Subject Classification (2000): 20E28, 20G40, 20C20 相似文献