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32
Subdivision Direction Selection In Interval Methods For Global Optimization
 SIAM J. Numer. Anal
, 1997
"... . The role of the interval subdivision selection rule is investigated in branchandbound algorithms for global optimization. The class of rules that allow convergence for the model algorithm is characterized, and it is shown that the four rules investigated satisfy the conditions of convergence. A ..."
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Cited by 46 (18 self)
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. The role of the interval subdivision selection rule is investigated in branchandbound algorithms for global optimization. The class of rules that allow convergence for the model algorithm is characterized, and it is shown that the four rules investigated satisfy the conditions of convergence. A numerical study with a wide spectrum of test problems indicates that there are substantial differences between the rules in terms of the required CPU time, the number of function and derivative evaluations and space complexity, and two rules can provide substantial improvements in efficiency. Key words. global optimization, interval arithmetic, interval subdivision AMS subject classifications. 65K05, 90C30 Abbreviated title: Subdivision directions in interval methods. 1. Introduction. Interval subdivision methods for global optimization [7, 21] aim at providing reliable solutions to global optimization problems min x2X f(x) (1) where the objective function f : IR n ! IR is continuo...
A Review Of Techniques In The Verified Solution Of Constrained Global Optimization Problems
, 1996
"... Elements and techniques of stateoftheart automatically verified constrained global optimization algorithms are reviewed, including a description of ways of rigorously verifying feasibility for equality constraints and a careful consideration of the role of active inequality constraints. Previousl ..."
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Cited by 25 (6 self)
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Elements and techniques of stateoftheart automatically verified constrained global optimization algorithms are reviewed, including a description of ways of rigorously verifying feasibility for equality constraints and a careful consideration of the role of active inequality constraints. Previously developed algorithms and general work on the subject are also listed. Limitations of present knowledge are mentioned, and advice is given on which techniques to use in various contexts. Applications are discussed. 1 INTRODUCTION, BASIC IDEAS AND LITERATURE We consider the constrained global optimization problem minimize OE(X) subject to c i (X) = 0; i = 1; : : : ; m (1.1) a i j x i j b i j ; j = 1; : : : ; q; where X = (x 1 ; : : : ; xn ) T . A general constrained optimization problem, including inequality constraints g(X) 0 can be put into this form by introducing slack variables s, replacing by s + g(X) = 0, and appending the bound constraint 0 s ! 1; see x2.2. 2 Chapter 1 W...
Interval Analysis for Guaranteed Nonlinear Parameter and State Estimation
"... This paper presents some tools based on interval analysis for guaranteed nonlinear parameter and state estimation in a boundederror context. These tools make it possible to compute outer (and sometimes inner) approximations of the set of all parameter or state vectors that are consistent with the m ..."
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Cited by 12 (5 self)
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This paper presents some tools based on interval analysis for guaranteed nonlinear parameter and state estimation in a boundederror context. These tools make it possible to compute outer (and sometimes inner) approximations of the set of all parameter or state vectors that are consistent with the model structure, measurements and noise bounds.
A comparison of some methods for solving linear interval equations
 SIAM Journal of Numerical Analysis
, 1997
"... Abstract. Certain cases in which the interval hull of a system of linear interval equations can be computed inexpensively are outlined. We extend a proposed technique of Hansen and Rohn with a formula that bounds the solution set of a system of equations whose coefficient matrix A = [A, A] is an Hm ..."
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Cited by 11 (0 self)
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Abstract. Certain cases in which the interval hull of a system of linear interval equations can be computed inexpensively are outlined. We extend a proposed technique of Hansen and Rohn with a formula that bounds the solution set of a system of equations whose coefficient matrix A = [A, A] is an Hmatrix; when A is centered about a diagonal matrix, these bounds are the smallest possible (i.e. the bounds are then the solution hull). Hansen’s scheme also computes the solution hull when the linear interval system Ax = b = [b, b] is such that A is inverse positive and b = −b � = 0. Earlier results of others also imply that, when A is an Mmatrix and b ≥ 0,b ≤ 0 or 0 ∈ b, interval Gaussian elimination (IGA) computes the hull. We also give a method of computing the solution hull inexpensively in many instances when A is inverse positive, given an outer approximation such as that obtained from IGA. Examples are used to compare these schemes under various conditions. Key words. numerical linear algebra, interval computations, inversepositive matrix, Hmatrix, interval hull, bounding solution sets
Guaranteed Error Bounds for Ordinary Differential Equations
 In Theory of Numerics in Ordinary and Partial Differential Equations
, 1994
"... Hamming once said, "The purpose of computing is insight, not numbers." If that is so, then the speed of our computers should be measured in insights per year, not operations per second. One key insight we wish from nearly all computing in engineering and scientific applications is, "How accurate is ..."
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Cited by 10 (0 self)
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Hamming once said, "The purpose of computing is insight, not numbers." If that is so, then the speed of our computers should be measured in insights per year, not operations per second. One key insight we wish from nearly all computing in engineering and scientific applications is, "How accurate is the answer?" Standard numerical analysis has developed techniques of forward and backward error analysis to help provide this insight, but even the best codes for computing approximate answers can be fooled. In contrast, validated computation ffl checks that the hypotheses of appropriate existence and uniqueness theorems are satisfied, ffl uses interval arithmetic with directed rounding to capture truncation and rounding errors in computation, and ffl organizes the computations to obtain as tight an enclosure of the answer as possible. These notes for a series of lectures at the VIth SERC Numerical Analysis Summer School, Leicester University, apply the principles of validated computatio...
Multisection in Interval BranchandBound Methods for Global Optimization II. Numerical Tests
, 1999
"... We have investigated variants of interval branchandbound algorithms for global optimization where the bisection step was substituted by the subdivision of the current, actual interval into many subintervals in a single iteration step. The results are published in two papers, the first one contains ..."
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Cited by 9 (2 self)
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We have investigated variants of interval branchandbound algorithms for global optimization where the bisection step was substituted by the subdivision of the current, actual interval into many subintervals in a single iteration step. The results are published in two papers, the first one contains the theoretical investigations on the convergence properties. An extensive numerical study indicates that multisection can substantially improve the efficiency of interval global optimization procedures, and multisection seems to be indispensable in solving hard global optimization problems in a reliable way.
Test Results for an Interval Branch and Bound Algorithm for EqualityConstrained Optimization
 In: Computational Methods and Applications, Kluwer
, 1995
"... . Various techniques have been proposed for incorporating constraints in interval branch and bound algorithms for global optimization. However, few reports of practical experience with these techniques have appeared to date. Such experimental results appear here. The underlying implementation includ ..."
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Cited by 8 (1 self)
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. Various techniques have been proposed for incorporating constraints in interval branch and bound algorithms for global optimization. However, few reports of practical experience with these techniques have appeared to date. Such experimental results appear here. The underlying implementation includes use of an approximate optimizer combined with a careful tesselation process and rigorous verification of feasibility. The experiments include comparison of methods of handling bound constraints and comparison of two methods for normalizing Lagrange multipliers. Selected test problems from the Floudas / Pardalos monograph are used, as well as selected unconstrained test problems appearing in reports of interval branch and bound methods for unconstrained global optimization. Keywords: constrained global optimization, verified computations, interval computations, bound constraints, experimental results 1. Introduction We consider the constrained global optimization problem minimize OE(X) s...
Controlling the ShortRange Order and Packing Densities of ManyParticle Systems
 J. Phys. Chem. B
, 2002
"... This paper explores the geometric availability of amorphous manyparticle configurations that conform to a given pair correlation function g(r). Such a study is required to observe the basic constraints of nonnegativity for g(r) as well as for its structure factor S(k). The hard sphere case receive ..."
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Cited by 8 (8 self)
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This paper explores the geometric availability of amorphous manyparticle configurations that conform to a given pair correlation function g(r). Such a study is required to observe the basic constraints of nonnegativity for g(r) as well as for its structure factor S(k). The hard sphere case receives special attention, to help identify what qualitative features play significant roles in determining upper limits to maximum amorphous packing densities. For that purpose, a fiveparameter test family of g's has been considered, which incorporates the known features of core exclusion, contact pairs, and damped oscillatory shortrange order beyond contact. Numerical optimization over this fiveparameter set produces a maximumpacking value for the fraction of covered volume, and about 5.8 for the mean contact number, both of which are within the range of previous experimental and simulational packing results. However, the corresponding maximumdensity g(r) and S(k) display some unexpected characteristics. These include absence of any pairs at about 1.4 times the sphere collision diameter, and a surprisingly large magnitude for S(k)0), the measure of macroscopicdistancescale density variations. On the basis of these results, we conclude that restoration of more subtle features to the testfunction family of g's (i.e., a split second peak, and a jump discontinuity at twice the collision diameter) will remove these unusual characteristics, while presumably increasing the maximum density slightly. A byproduct of our investigation is a lower bound on the maximum density for random sphere packings in d dimensions, which is sharper than a wellknown lower bound for regular lattice packings for d g 3
Existence Verification For Higher Degree Singular Zeros Of Complex Nonlinear Systems
 SIAM J. Number. Anal
, 2000
"... It is known that, in general, no computational techniques can verify the existence of a singular solution of the nonlinear system of n equations in n variables within a given region x of nspace. However, computational verication that a given number of true solutions exist within a region in comple ..."
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Cited by 8 (3 self)
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It is known that, in general, no computational techniques can verify the existence of a singular solution of the nonlinear system of n equations in n variables within a given region x of nspace. However, computational verication that a given number of true solutions exist within a region in complex space containing x is possible. That can be done by computation of the topological degree. In a previous paper, we presented theory and algorithms for the simplest case, when the rankdefect of the Jacobi matrix at the solution is one and the topological index is 2. Here, we will generalize that result to arbitrary topological index d 2: We present theory, algorithms, and experimental results. We also present a heuristic for determining the degree, obtaining a value that we can subsequently verify with our algorithms. Key words. complex nonlinear systems, interval computations, veried computations, singularities, topological degree AMS subject classications. 65G10, 65H10 1.
Existence Verification For Singular Zeros Of Complex Nonlinear Systems
 SIAM J. Numer. Anal
, 2000
"... Computational fixed point theorems can be used to automatically verify existence and uniqueness of a solution to a nonlinear system of n equations in n variables ranging within a given region of nspace. Such computations succeed, however, only when the Jacobi matrix is nonsingular everywhere in thi ..."
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Cited by 8 (4 self)
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Computational fixed point theorems can be used to automatically verify existence and uniqueness of a solution to a nonlinear system of n equations in n variables ranging within a given region of nspace. Such computations succeed, however, only when the Jacobi matrix is nonsingular everywhere in this region. However, in problems such as bifurcation problems or surface intersection problems, the Jacobi matrix can be singular, or nearly so, at the solution. For n real variables, when the Jacobi matrix is singular, tiny perturbations of the problem can result in problems either with no solution in the region, or with more than one; thus no general computational technique can prove existence and uniqueness. However, for systems of n complex variables, the multiplicity of such a solution can be verified. That is the subject of this paper. Such verification is possible by computing the topological degree, but such computations heretofore have required a global search on the (n  1)dimensional boundary of an ndimensional region. Here it is observed that preconditioning leads to a system of equations whose topological degree can be computed with a much lowerdimensional search. Formulas are given for this computation, and the special case of rankdefect one is studied, both theoretically and empirically. Verification is possible for certain subcases of the real case. That will be the subject of a companion paper. Key words. complex nonlinear systems, interval computations, verified computations, singularities, topological degree AMS subject classifications. 65G10, 65H10 PII. S0036142999361074 1.