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1.
Reuven Rubinstein 《Methodology and Computing in Applied Probability》2009,11(4):491-549
We present a randomized algorithm, called the cloning algorithm, for approximating the solutions of quite general NP-hard
combinatorial optimization problems, counting, rare-event estimation and uniform sampling on complex regions. Similar to the
algorithms of Diaconis–Holmes–Ross and Botev–Kroese the cloning algorithm is based on the MCMC (Gibbs) sampler equipped with
an importance sampling pdf and, as usual for randomized algorithms, it uses a sequential sampling plan to decompose a “difficult”
problem into a sequence of “easy” ones. The cloning algorithm combines the best features of the Diaconis–Holmes–Ross and the
Botev–Kroese. In addition to some other enhancements, it has a special mechanism, called the “cloning” device, which makes
the cloning algorithm, also called the Gibbs cloner fast and accurate. We believe that it is the fastest and the most accurate randomized algorithm for counting known so far.
In addition it is well suited for solving problems associated with the Boltzmann distribution, like estimating the partition
functions in an Ising model. We also present a combined version of the cloning and cross-entropy (CE) algorithms. We prove
the polynomial complexity of a particular version of the Gibbs cloner for counting. We finally present efficient numerical
results with the Gibbs cloner and the combined version, while solving quite general integer and combinatorial optimization
problems as well as counting ones, like SAT. 相似文献
2.
Yu. V. Matiyasevich 《Proceedings of the Steklov Institute of Mathematics》2011,275(1):118-132
This survey presents various theorems (obtained mainly by specialists in mathematical logic and computability theory) stating
the impossibility of algorithms for solving certain Diophantine problems. Often the technique developed for obtaining such
“negative” results also allows one to prove many “positive” theorems on the possibility of formulating Diophantine problems
with special properties. This survey also lists a number of questions that remain open. 相似文献
3.
Gábor Czédli 《Mathematica Slovaca》2011,61(6):859-870
For each of the relations “less than or equal to”, “less than”, “covered by”, and “covered by or equal to”, we characterize
finite orders (also called posets) with the property that the pair of Galois closure operators induced by the relation in
question coincides with the pair of closure operators introduced and applied in our previous paper in 2007. We also consider
the “less than or equal to” relation between the set of join-irreducible elements and the set of meet-irreducible elements,
and we show that the above-mentioned pairs of closure operators coincide for finite modular lattices. 相似文献
4.
Roland and Varadhan (Appl. Numer. Math., 55:215–226, 2005) presented a new idea called “squaring” to improve the convergence of Lemaréchal’s scheme for solving nonlinear fixed-point
problems. Varadhan and Roland (Squared extrapolation methods: A new class of simple and efficient numerical schemes for accelerating
the convergence of the EM algorithm, Department of Biostatistics Working Paper. Johns Hopkins University, , 2004) noted that Lemaréchal’s scheme can be viewed as a member of the class of polynomial extrapolation methods with cycling that
uses two fixed-point iterations per cycle. Here we combine these two ideas, cycled extrapolation and squaring, and construct
a new class of methods, called squared polynomial methods (SQUAREM), for accelerating the convergence of fixed-point iterations.
Our main goal is to evaluate whether the squaring device is effective in improving the rate of convergence of cycled extrapolation
methods that use more than two fixed-point iterations per cycle. We study the behavior of the new schemes on an image reconstruction
problem for positron emission tomography (PET) using simulated data. Our numerical experiments show the effectiveness of first-
and higher-order squared polynomial extrapolation methods in accelerating image reconstruction, and also their relative superiority
compared to the classical, “unsquared” vector polynomial methods. 相似文献
5.
In this paper, we present efficient solution approaches for discrete multi-facility competitive interaction model. Applying
the concept of “Tangent Line Approximation” presented by the authors in their previous work, we develop efficient computational
approaches—both exact and approximate (with controllable error bound α). Computational experiments show that the approximate approach (with small α) performs extremely well solving large scale problems while the exact approach performs very well for small to medium-sized
problems. 相似文献
6.
Tanabe (1988) proposed a variation of the classical Newton method for solving nonlinear systems of equations, the so-called Centered Newton
method. His idea was based on a deviation of the Newton direction towards a variety called “Central Variety”. In this paper
we prove that the Centered Newton method is locally convergent and we present a globally convergent method based on the centered
direction used by Tanabe. We show the effectiveness of our proposal for solving nonlinear system of equations and compare
with the Newton method with line search. 相似文献
7.
C. Bourdarias 《Numerische Mathematik》2001,87(4):645-662
Summary. The “fluctuation-splitting schemes” (FSS in short) have been introduced by Roe and Sildikover to solve advection equations
on rectangular grids and then extended to triangular grids by Roe, Deconinck, Struij... For a two dimensional nonlinear scalar
conservation law, we consider the case of a triangular grid and of a kinetic approach to reduce the discretization of the
nonlinear equation to a linear equation and apply a particular FSS called N-scheme. We show that the resulting scheme converges
strongly in in a finite volume sense.
Received February 25, 1997 / Revised version received November 8, 1999 / Published online August 24, 2000 相似文献
8.
Representations of solutions of boundary value problems for simple domains in the Monte Carlo algorithms are widely distributed
[2]. In particular, widespread use is made of such a representation for the ball. It allows one to formally write an integral
equation of the second kind for the required function in an arbitrary domain with regular boundary. Moreover, with the involvement
of the joining conditions [1], one can picture a possible construction of a random process to “solve” the problem. However,
the “walk in spheres” process, which solves the first boundary value problem for the Poisson equation, results in ɛ-biased estimators, and so the introduction of a regularization parameter is required.
The authors investigate in detail the “walk in hemispheres” method, which has been proposed earlier by A. S. Sipin [10] without
full justification. The use of the Green’s function for the hemisphere makes it possible to obtain estimators for the first
and the third boundary value problems, as well as for the problem with discontinuous derivative; for a broad class of domains,
these estimators are shown to be unbiased. The algorithms proposed feature a high degree of parallelism. Results of solving
model problems are presented. 相似文献
9.
Kalpana Dahiya Surjeet Kaur Suneja Vanita Verma 《Computational Optimization and Applications》2007,36(1):67-82
In this paper a minimization problem with convex objective function subject to a separable convex inequality constraint “≤”
and bounded variables (box constraints) is considered. We propose an iterative algorithm for solving this problem based on
line search and convergence of this algorithm is proved. At each iteration, a separable convex programming problem with the
same constraint set is solved using Karush-Kuhn-Tucker conditions. Convex minimization problems subject to linear equality/
linear inequality “≥” constraint and bounds on the variables are also considered. Numerical illustration is included in support
of theory. 相似文献
10.
Edward M. Bolger 《International Journal of Game Theory》2000,29(1):93-99
In Bolger [1993], an efficient value was obtained for a class of games called games with n players and r alternatives. In these games, each of the n players must choose one and only one of the r alternatives. This value can be used to determine a player’s “a priori” value in such a game. In this paper, we show that
the value has a consistency property similar to the “consistency” for TU games in Hart/Mas-Colell [1989] and we present a
set of axioms (including consistency) which characterizes this value.
The games considered in this paper differ from the multi-choice games considered by Hsiao and Raghavan [1993]. They consider
games in which the actions of the players are ordered in the sense that, if i >j, then action i carries more “weight” than action j.
These games also differ from partition function games in that the worth of a coalition depends not only on the partitioning
of the players but also on the action chosen by each subset of the partition.
Received: April 1994/final version: June 1999 相似文献
11.
Franz Wirl 《Computational Management Science》2008,5(4):393-401
This note shows that the second derivative of the value function exists (across a stopping threshold, short “super contact”)
if reversibly stopping and entering involves no cost, called “switching”. This holds for discrete (real option) as well as
for continuous stochastic control problems and proves particularly suitable in real option set ups since it provides the lacking
boundary condition. However, super contact does not hold in dynamic games. A simple example documents the applicability of
this condition.
This paper was written during my visit of the University of Technology, Sydney (UTS) and I am grateful for the hospitality
of and the stimulus at the School of Finance and Economics, in particular to Carl Chiarella. I also acknowledge many helpful
discussions with Thomas Dangl on related issues, valuable suggestions from a referee and last but not least encouragement
by Josef Kallrath 相似文献
12.
T. V. Panchapagesan 《Rendiconti del Circolo Matematico di Palermo》1995,44(3):417-440
The concept of an orthogonal spectral representation (OTSR) of a Hilbert spaceH relative to a spectral measureE(.) is introduced and it is shown that every Hilbert space admits an OTSR relative to a given spectral measure. Apart from
the various results obtained about OTSRs, the principal result of Allan Brown (1974) is deduced as an easy consequence of
this study. A new complete system of unitary invariants called the “equivalence of OTSRs”, is given for spectral measures.
Two special types of OTSRs called “BOTSR” and “COBOTSR” are introduced and characterized respectively in terms of the “GCGS-property”
and “CGS-property” of the associated spectral measure. Various complete systems of unitary invariants are given for spectral
measures with the GCGS-property. Finally, the Wecken-Plesner-Rohlin theorem on hermitian operators with simple spectra is
generalized to arbitrary spectral measures. 相似文献
13.
E. A. Al-Said A. H. Almualim M. A. Noor 《Journal of Optimization Theory and Applications》2010,146(3):810-812
In this article some comments on the paper “parametric cubic spline approach to the solution of a system of second order boundary
value problems” in (Khan and Aziz, J. Optim. Theory Appl. 118:45–54, 2003) are given. This paper concerns with a numerical method for solving a second order boundary value problem associated with
obstacle, unilateral and contact problems. Corrections are given for the convergence analysis of the numerical method and
the computational experiments. 相似文献
14.
The idea of a finite collection of closed sets having “linearly regular intersection” at a point is crucial in variational
analysis. This central theoretical condition also has striking algorithmic consequences: in the case of two sets, one of which
satisfies a further regularity condition (convexity or smoothness, for example), we prove that von Neumann’s method of “alternating
projections” converges locally to a point in the intersection, at a linear rate associated with a modulus of regularity. As
a consequence, in the case of several arbitrary closed sets having linearly regular intersection at some point, the method
of “averaged projections” converges locally at a linear rate to a point in the intersection. Inexact versions of both algorithms
also converge linearly.
Research of A.S. Lewis supported in part by National Science Foundation Grant DMS-0504032.
Research of D.R. Luke supported in part by National Science Foundation Grant DMS-0712796. 相似文献
15.
A puzzle called “M
13” J. H. Conway has described recently is explained. We report an implementation of the puzzle in the programming language
Java. The program allows the human user to “play M
13” interactively (and to cheat by solving it automatically). The program is an example on how to bring to life a nice piece
of discrete mathematics. In this sense it presents not only a didactical way of seeing “mathematics at work”, but also displays
the stabilizer chain method developed by C. Sims to solve group theoretic puzzles, the most famous of which being Rubik's
cube. 相似文献
16.
We study the properties of the ergosurface of the Pomeransky–Senkov black rings, and show that it splits into an “inner” and
an “outer” region. As for the singular set, the topology of the “outer ergosurface” depends upon the value of parameters. 相似文献
17.
Typically local search methods for solving constraint satisfaction problems such as GSAT, WalkSAT, DLM, and ESG treat the
problem as an optimisation problem. Each constraint contributes part of a penalty function in assessing trial valuations.
Local search examines the neighbours of the current valuation, using the penalty function to determine a “better” neighbour
valuation to move to, until finally a solution which satisfies all the constraints is found. In this paper we investigate
using some of the constraints as “hard” constraints, that are always satisfied by every trial valuation visited, rather than
as part of a penalty function. In this way these constraints reduce the possible neighbours in each move and also the overall
search space. The treating of some constraints as hard requires that the space of valuations that are satisfied is “connected”
in order to guarantee that a solution can be found from any starting position within the region; thus the concept of islands
and the name “island confinement method” arises. Treating some constraints as hard provides new difficulties for the search
mechanism since the search space becomes more jagged, and there are more deep local minima. A new escape strategy is needed.
To demonstrate the feasibility and generality of our approach, we show how the island confinement method can be incorporated
in, and significantly improve, the search performance of two successful local search procedures, DLM and ESG, on SAT problems
arising from binary CSPs.
A preliminary version of this paper appeared in AAAI’2002. 相似文献
18.
Christian Meyer Arnd Rösch Fredi Tröltzsch 《Computational Optimization and Applications》2006,33(2-3):209-228
This paper addresses the regularization of pointwise state constraints in optimal control problems. By analyzing the associated
dual problem, it is shown that the regularized problems admit Lagrange multipliers in L2-spaces. Under a certain boundedness assumption, the solution of the regularized problem converges to the one of the original
state constrained problem. The results of our analysis are confirmed by numerical tests.
Supported by the DFG Research Center “Mathematics for key technologies” (FZT 86) in Berlin. 相似文献
19.
We prove strong convergence of the viscosity approximation process for nonexpansive nonself multimaps. Furthermore, an explicit iteration process which converges strongly to a fixed point of multimap T is constructed. It is worth mentioning that, unlike other authors, we do not impose the condition "Tz = {z}" on the map T. 相似文献
20.
Laurent Stolovitch 《Publications Mathématiques de L'IHéS》2005,102(1):99-165
Let X be a germ of holomorphic vector field at the origin of Cn and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system.
The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first
integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets
in a neighborhood of the origin. These are biholomorphic to the intersection of a polydisc with an analytic set of the form
“resonant monomials = constants”. Such a biholomorphism conjugates the restriction of X to one of its invariant varieties
to the restriction of a linear diagonal vector field to a toric variety. Moreover, we show that the set of “frequencies” defining
the invariant sets is of positive measure. 相似文献