共查询到20条相似文献,搜索用时 312 毫秒
1.
Scenario reduction in stochastic programming 总被引:2,自引:0,他引:2
Given a convex stochastic programming problem with a discrete initial probability distribution, the problem of optimal scenario
reduction is stated as follows: Determine a scenario subset of prescribed cardinality and a probability measure based on this
set that is the closest to the initial distribution in terms of a natural (or canonical) probability metric. Arguments from
stability analysis indicate that Fortet-Mourier type probability metrics may serve as such canonical metrics. Efficient algorithms
are developed that determine optimal reduced measures approximately. Numerical experience is reported for reductions of electrical
load scenario trees for power management under uncertainty. For instance, it turns out that after 50% reduction of the scenario
tree the optimal reduced tree still has about 90% relative accuracy.
Received: July 2000 / Accepted: May 2002 Published online: February 14, 2003
Key words. stochastic programming – quantitative stability – Fortet-Mourier metrics – scenario reduction – transportation problem –
electrical load scenario tree
Mathematics Subject Classification (1991): 90C15, 90C31 相似文献
2.
Andrzej Ruszczyński 《Mathematical Programming》2002,93(2):195-215
We consider stochastic programming problems with probabilistic constraints involving random variables with discrete distributions.
They can be reformulated as large scale mixed integer programming problems with knapsack constraints. Using specific properties
of stochastic programming problems and bounds on the probability of the union of events we develop new valid inequalities
for these mixed integer programming problems. We also develop methods for lifting these inequalities. These procedures are
used in a general iterative algorithm for solving probabilistically constrained problems. The results are illustrated with
a numerical example.
Received: October 8, 2000 / Accepted: August 13, 2002 Published online: September 27, 2002
Key words. stochastic programming – integer programming – valid inequalities 相似文献
3.
In this paper we address a general Goal Programming problem with linear objectives, convex constraints, and an arbitrary
componentwise nondecreasing norm to aggregate deviations with respect to targets. In particular, classical Linear Goal Programming
problems, as well as several models in Location and Regression Analysis are modeled within this framework.
In spite of its generality, this problem can be analyzed from a geometrical and a computational viewpoint, and a unified solution
methodology can be given. Indeed, a dual is derived, enabling us to describe the set of optimal solutions geometrically. Moreover,
Interior-Point methods are described which yield an $\varepsilon$-optimal solution in polynomial time.
Received: February 1999 / Accepted: March 2002 Published online: September 5, 2002
Key words. Goal programming – closest points – interior point methods – location – regression 相似文献
4.
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
5.
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
6.
We give a policy-improvement type algorithm to locate an optimal pure stationary strategy for discounted stochastic games
with perfect information. A graph theoretic motivation for our algorithm is presented as well.
Received: January 1998 / Accepted: May 2002 Published online: February 14, 2003
Key words. stochastic games – MDP – perfect information – policy iteration
Partially Funded by NSF Grant DMS 930-1052 and DMS 970-4951 相似文献
7.
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
8.
We study Graver test sets for linear two-stage stochastic integer programs and show that test sets can be decomposed into
finitely many building blocks whose number is independent on the number of scenarios of the stochastic program. We present
a finite algorithm to compute the building blocks directly, without prior knowledge of test set vectors. Once computed, building
blocks can be employed to solve the stochastic program by a simple augmentation scheme, again without explicit knowledge of
test set vectors. Finally, we report preliminary computational experience.
Received: March 14, 2002 / Accepted: March 27, 2002 Published online: September 27, 2002
Key words. test sets – stochastic integer programming – decomposition methods
Mathematics Subject Classification (2000): 90C15, 90C10, 13P10 相似文献
9.
We define a convex extension of a lower semi-continuous function to be a convex function that is identical to the given function
over a pre-specified subset of its domain. Convex extensions are not necessarily constructible or unique. We identify conditions
under which a convex extension can be constructed. When multiple convex extensions exist, we characterize the tightest convex
extension in a well-defined sense. Using the notion of a generating set, we establish conditions under which the tightest
convex extension is the convex envelope. Then, we employ convex extensions to develop a constructive technique for deriving
convex envelopes of nonlinear functions. Finally, using the theory of convex extensions we characterize the precise gaps exhibited
by various underestimators of $x/y$ over a rectangle and prove that the extensions theory provides convex relaxations that
are much tighter than the relaxation provided by the classical outer-linearization of bilinear terms.
Received: December 2000 / Accepted: May 2002 Published online: September 5, 2002
RID="*"
ID="*" The research was funded in part by a Computational Science and Engineering Fellowship to M.T., and NSF CAREER award
(DMI 95-02722) and NSF/Lucent Technologies Industrial Ecology Fellowship (NSF award BES 98-73586) to N.V.S.
Key words. convex hulls and envelopes – multilinear functions – disjunctive programming – global optimization 相似文献
10.
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
11.
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
12.
This paper presents a renormalization and homogenization theory for fractional-in-space or in-time diffusion equations with
singular random initial conditions. The spectral representations for the solutions of these equations are provided. Gaussian
and non-Gaussian limiting distributions of the renormalized solutions of these equations are then described in terms of multiple
stochastic integral representations.
Received: 30 May 2000 / Revised version: 9 November 2001 / Published online: 10 September 2002
Mathematics Subject Classification (2000): Primary 62M40, 62M15; Secondary 60H05, 60G60
Key words or phrases: Fractional diffusion equation – Scaling laws – Renormalised solution – Long-range dependence – Non-Gaussian scenario – Mittag-Leffler
function – Stable distributions – Bessel potential – Riesz potential 相似文献
13.
14.
We study a general multiobjective optimization problem with variational inequality, equality, inequality and abstract constraints.
Fritz John type necessary optimality conditions involving Mordukhovich coderivatives are derived. They lead to Kuhn-Tucker
type necessary optimality conditions under additional constraint qualifications including the calmness condition, the error
bound constraint qualification, the no nonzero abnormal multiplier constraint qualification, the generalized Mangasarian-Fromovitz
constraint qualification, the strong regularity constraint qualification and the linear constraint qualification. We then
apply these results to the multiobjective optimization problem with complementarity constraints and the multiobjective bilevel
programming problem.
Received: November 2000 / Accepted: October 2001 Published online: December 19, 2002
Key Words. Multiobjective optimization – Variational inequality – Complementarity constraint – Constraint qualification – Bilevel programming
problem – Preference – Utility function – Subdifferential calculus – Variational principle
Research of this paper was supported by NSERC and a University of Victoria Internal Research Grant
Research was supported by the National Science Foundation under grants DMS-9704203 and DMS-0102496
Mathematics Subject Classification (2000): Sub49K24, 90C29 相似文献
15.
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
16.
Minimizing risk models in stochastic shortest path problems 总被引:1,自引:0,他引:1
Yoshio Ohtsubo 《Mathematical Methods of Operations Research》2003,57(1):79-88
17.
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
18.
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
19.
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
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