Interval Methods

Ratschek, Helmut; Rokne, Jon G.

doi:10.1007/978-1-4615-2025-2_14

Cited by 38 publications

(30 citation statements)

References 33 publications

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“…This is in line with the conclusions by Goualard [4]. For those problems having a good transversal (2,3,4,9,10,12,13), the results of the heuristics vary widely, and some problems cannot be solved in the allotted time. At this point, the only reason that comes to mind is that the heuristics did choose a bad transversal and were stuck with it since the choice is not reconsidered dynamically.…”

Section: Fig 1 Comparison Of Un and Bcsupporting

confidence: 88%

“…Many numerical methods have been extended to interval arithmetic [11,13]. Given a system of nonlinear equations of the form:…”

Section: An Interval Nonlinear Gauss-seidel Methodsmentioning

confidence: 99%

“…The extensions to interval arithmetic [10] of Newton and nonlinear Gauss-Seidel methods [13] do not suffer from lack of convergence or loss of solutions that cripple their floating-point counterparts, which makes them well suited to solve systems of highly nonlinear equations.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Selection of a Transversal to Solve Nonlinear Systems with Interval Arithmetic

Goualard

Jermann

2006

Computational Science – ICCS 2006

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Abstract. This paper investigates the impact of the selection of a transversal on the speed of convergence of interval methods based on the nonlinear Gauss-Seidel scheme to solve nonlinear systems of equations. It is shown that, in a marked contrast with the linear case, such a selection does not speed up the computation in the general case; directions for researches on more flexible methods to select projections are then discussed.

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Section: Fig 1 Comparison Of Un and Bcsupporting

confidence: 88%

“…Many numerical methods have been extended to interval arithmetic [11,13]. Given a system of nonlinear equations of the form:…”

Section: An Interval Nonlinear Gauss-seidel Methodsmentioning

confidence: 99%

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On the Selection of a Transversal to Solve Nonlinear Systems with Interval Arithmetic

Goualard

Jermann

2006

Computational Science – ICCS 2006

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“…Various parallel implementations of the B&B paradigm are well-known [8]. Our approach is closest to the work presented in [9] where the bounding technique is based on interval-arithmetic [10]. The important differences stem from the fact that our approach is targeted at the P2P network model described above, and it is based on gossip instead of shared memory.…”

mentioning

confidence: 99%

“…The algorithm that is run at all nodes is shown in Algorithm 1. The lower bound for an interval is calculated using interval arithmetic [10], which guarantees that the calculated bound is indeed a lower bound. We start the algorithm by sending the search domain D with lower bound b = ∞ to a random node.…”

mentioning

confidence: 99%

Peer-to-Peer Optimization in Large Unreliable Networks with Branch-and-Bound and Particle Swarms

Bánhelyi

Biazzini

Montresor

et al. 2009

Lecture Notes in Computer Science

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Abstract. Decentralized peer-to-peer (P2P) networks (lacking a GRID-style resource management and scheduling infrastructure) are an increasingly important computing platform. So far, little is known about the scaling and reliability of optimization algorithms in P2P environments. In this paper we present empirical results comparing two P2P algorithms for real-valued search spaces in large-scale and unreliable networks. Some interesting, and perhaps counter-intuitive findings are presented: for example, failures in the network can in fact significantly improve performance under some conditions. The two algorithms that are compared are a known distributed particle swarm optimization (PSO) algorithm and a novel P2P branch-and-bound (B&B) algorithm based on interval arithmetic. Although our B&B algorithm is not a black-box heuristic, the PSO algorithm is competitive in certain cases, in particular, in larger networks. Comparing two rather different paradigms for solving the same problem gives a better characterization of the limits and possibilities of optimization in P2P networks.

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Optimization Methods

Lewis¹,

Trosset²

2005

Encyclopedia of Statistics in Behavioral Science

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Optimization is concerned with identifying the best element in a set of competing possibilities. Except in special circumstances, one must use numerical methods to search for solutions iteratively. If one's problem is well‐behaved, for example, the objective to be minimized is smooth and the feasible possibilities are smoothly defined by finite numbers of equality and inequality constraints, then one can use a number of sophisticated methods to identify local solutions, that is, possibilities that are as good or better than any other nearby possibilities. However, finding global solutions is much harder.

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Interval Methods

Cited by 38 publications

References 33 publications

On the Selection of a Transversal to Solve Nonlinear Systems with Interval Arithmetic

On the Selection of a Transversal to Solve Nonlinear Systems with Interval Arithmetic

Peer-to-Peer Optimization in Large Unreliable Networks with Branch-and-Bound and Particle Swarms

Optimization Methods

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