Results 1  10
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21
Intlab  Interval Laboratory
"... . INTLAB is a Matlab toolbox supporting real and complex interval scalars, vectors, and matrices, as well as sparse real and complex interval matrices. It is designed to be very fast. In fact, it is not much slower than the fastest pure floating point algorithms using the fastest compilers available ..."
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Cited by 57 (2 self)
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. INTLAB is a Matlab toolbox supporting real and complex interval scalars, vectors, and matrices, as well as sparse real and complex interval matrices. It is designed to be very fast. In fact, it is not much slower than the fastest pure floating point algorithms using the fastest compilers available (the latter, of course, without verification of the result). Portability is assured by implementing all algorithms in Matlab itself with exception of exactly three routines for switching the rounding downwards, upwards and to nearest. Timing comparisons show that the used concept achieves the anticipated speed with identical code on a variety of computers, ranging from PC's to parallel computers. INTLAB may be freely copied from our home page. 1. Introduction. The INTLAB concept splits into two parts. First, a new concept of a fast interval library is introduced. The main advantage (and difference to existing interval libraries) is that identical code can be used on a variety of computer a...
A Global Optimization Method, αBB, for General TwiceDifferentiable Constrained NLPs: I  Theoretical Advances
, 1997
"... In this paper, the deterministic global optimization algorithm, αBB, (αbased Branch and Bound) is presented. This algorithm offers mathematical guarantees for convergence to a point arbitrarily close to the global minimum for the large class of twicedifferentiable NLPs. The key idea is the constru ..."
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Cited by 51 (3 self)
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In this paper, the deterministic global optimization algorithm, αBB, (αbased Branch and Bound) is presented. This algorithm offers mathematical guarantees for convergence to a point arbitrarily close to the global minimum for the large class of twicedifferentiable NLPs. The key idea is the construction of a converging sequence of upper and lower bounds on the global minimum through the convex relaxation of the original problem. This relaxation is obtained by (i) replacing all nonconvex terms of special structure (i.e., bilinear, trilinear, fractional, fractional trilinear, univariate concave) with customized tight convex lower bounding functions and (ii) by utilizing some α parameters as defined by Maranas and Floudas (1994b) to generate valid convex underestimators for nonconvex terms of generic structure. In most cases, the calculation of appropriate values for the α parameters is a challenging task. A number of approaches are proposed, which rigorously generate a set of α par...
Fast And Parallel Interval Arithmetic
 BIT
"... . Inmumsupremum interval arithmetic is widely used because of ease of implementation and narrow results. In this note we show that the overestimation of midpointradius interval arithmetic compared to power set operations is uniformly bounded by a factor 1.5 in radius. This is true for the four bas ..."
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Cited by 25 (2 self)
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. Inmumsupremum interval arithmetic is widely used because of ease of implementation and narrow results. In this note we show that the overestimation of midpointradius interval arithmetic compared to power set operations is uniformly bounded by a factor 1.5 in radius. This is true for the four basic operations as well as for vector and matrix operations, over real and over complex numbers. Moreover, we describe an implementation of midpointradius interval arithmetic entirely using BLAS. Therefore, in particular, matrix operations are very fast on almost any computer, with minimal eort for the implementation. Especially, with the new denition it is seemingly the rst time that full advantage can be taken of the speed of vector and parallel architectures. The algorithms have been implemented in the Matlab interval toolbox INTLAB. Keywords. Interval arithmetic, parallel computer, BLAS, midpointradius, inmumsupremum, AMS subject classications. 65G10 1. Introduction and notati...
Bounds for linear recurrences with restricted coefficients
 Journal of Inequalities in Pure and Applied Mathematics 4, 2, Article
"... Abstract. This paper provides bounds for secondorder linear recurrences with restricted coefficients. It is determined that whenever the coefficients of the associated monic equation are less than the constant {l/3) 1 1 3, all solutions tend to zero at an exponential rate. This constant is optimal. ..."
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Cited by 13 (6 self)
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Abstract. This paper provides bounds for secondorder linear recurrences with restricted coefficients. It is determined that whenever the coefficients of the associated monic equation are less than the constant {l/3) 1 1 3, all solutions tend to zero at an exponential rate. This constant is optimal. Explicit inequalities are also provided, and some residue class structure is revealed. 1.
Lower Bound Functions for Polynomials
 J. Computational and Applied Mathematics
, 2003
"... This paper addresses the construction of relaxations for problems involving multivariate polynomials. The major goal is to show how nonconvex multivariate polynomial terms can be replaced by affine and convex lower bound functions which are computed by using Bernstein coefficients. These bound func ..."
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Cited by 10 (4 self)
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This paper addresses the construction of relaxations for problems involving multivariate polynomials. The major goal is to show how nonconvex multivariate polynomial terms can be replaced by affine and convex lower bound functions which are computed by using Bernstein coefficients. These bound functions may be used in any relaxation method described in the above literature, whenever these approaches do not deliver satisfactory results for polynomial terms of higher degree. Moreover, several properties of these bound functions are discussed. For properties of Bernstein polynomials the reader is referred to Cargo and Shisha [5], Farin [7], Garloff [11], Garloff, Jansson and Smith [12], and Zettler and Garloff [30]. By using Bernstein coefficients, bounds for the range of a multivariate polynomial over a box can be computed. It was shown by Stahl [28] that in the univariate case these bounds are often tighter than bounds which are obtained by applying interval computation techniques (cf. Neumaier [21], Ratschek and Rokne [23]). In [19] a method is presented by which piecewise linear lower (and equally linear upper) bound functions for multivariate polynomials can be obtained. This leads to tight enclosures of the given polynomials which are important, e.g., in intersection testing. The construction is presented there in detail in the univariate and bivariate cases. However, these lower bound functions are in general not convex. So the convex envelope of the piecewise linear lower bound functions has to be taken, requiring additional effort. The paper is organised as follows. In the next section some basic definitions and properties of Bernstein polynomials are given. Affine and convex lower bound functions based on the Bernstein expansion are presented in Section 3. An error bound for the...
A Global Optimization Method, αBB, for Process Design
 COMPUT. CHEM. ENG
, 1996
"... A global optimization algorithm, αBB, for twicedifferentiable NLPs is presented. It operates within a branchandbound framework and requires the construction of a convex lower bounding problem. A technique to generate such a valid convex underestimator for arbitrary twicedifferentiable functions ..."
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Cited by 7 (1 self)
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A global optimization algorithm, αBB, for twicedifferentiable NLPs is presented. It operates within a branchandbound framework and requires the construction of a convex lower bounding problem. A technique to generate such a valid convex underestimator for arbitrary twicedifferentiable functions is described. The αBB has been applied to a variety of problems and a summary of the results obtained is provided.
Phase Stability with Cubic Equations of State: A Global Optimization Approach
 AIChE J
, 2000
"... Calculation of phase and chemical equilibria is of fundamental importance for the design and simulation of chemical processes. Methods that minimize the Gibbs free energy provide equilibrium solutions that are only candidates for the true equilibrium solution. This is because the number and type of ..."
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Cited by 7 (0 self)
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Calculation of phase and chemical equilibria is of fundamental importance for the design and simulation of chemical processes. Methods that minimize the Gibbs free energy provide equilibrium solutions that are only candidates for the true equilibrium solution. This is because the number and type of phases must be assumed before the Gibbs energy minimization problem can be formulated. The tangent plane stability criterion is a means of determining the stability of a candidate equilibrium solution. The Gibbs energy minimization problem and the tangent plane stability problem are very challenging due to the highly nonlinear thermodynamic functions that are used. In this work the goal is to develop a global optimization approach for the tangent plane stability problem that (i) provides a theoretical guarantee about the stability of the candidate equilibrium solution and (ii) is computationally efficient. Cubic equations of state are used in this approach due to their ability to accurately ...
b4m  A free interval arithmetic toolbox for Matlab based on BIAS
"... this documentation. 4 The class interval ..."
Computational Experience with Rigorous Error Bounds for the Netlib Linear Programming Library
 Reliable Computing
, 2006
"... Abstract. The Netlib library of linear programming problems is a well known suite containing many real world applications. Recently it was shown by Ordóñez and Freund that 71 % of these problems are illconditioned. Hence, numerical difficulties may occur. Here, we present rigorous results for this ..."
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Cited by 3 (1 self)
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Abstract. The Netlib library of linear programming problems is a well known suite containing many real world applications. Recently it was shown by Ordóñez and Freund that 71 % of these problems are illconditioned. Hence, numerical difficulties may occur. Here, we present rigorous results for this library that are computed by a verification method using interval arithmetic. In addition to the original input data of these problems we also consider interval input data. The computed rigorous bounds and the performance of the algorithms are related to the distance to the next illposed linear programming problem.
Improving interval enclosures
, 2009
"... This paper serves as background information for the Vienna proposal for interval standardization, explaining what is needed in practice to make competent use of the interval arithmetic provided by an implementation of the standard to be. Discussed are methods to improve the quality of interval encl ..."
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Cited by 2 (0 self)
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This paper serves as background information for the Vienna proposal for interval standardization, explaining what is needed in practice to make competent use of the interval arithmetic provided by an implementation of the standard to be. Discussed are methods to improve the quality of interval enclosures of the range of a function over a box, considerations of possible hardware support facilitating the implementation of such methods, and the results of a simple interval challenge that I had posed to the reliable computing mailing list on November 26, 2008. Also given is an example of a bound constrained global optimization problem in 4 variables that has a 2dimensional continuum of global minimizers. This makes standard branch and bound codes extremely slow, and therefore may serve as a useful degenerate test problem.