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41
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 60 (20 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...
Empirical Evaluation Of Innovations In Interval Branch And Bound Algorithms For Nonlinear Systems
 SIAM J. Sci. Comput
, 1994
"... . Interval branch and bound algorithms for finding all roots use a combination of a computational existence / uniqueness procedure and a tesselation process (generalized bisection). Such algorithms identify, with mathematical rigor, a set of boxes that contains unique roots and a second set within w ..."
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Cited by 20 (10 self)
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. Interval branch and bound algorithms for finding all roots use a combination of a computational existence / uniqueness procedure and a tesselation process (generalized bisection). Such algorithms identify, with mathematical rigor, a set of boxes that contains unique roots and a second set within which all remaining roots must lie. Though each root is contained in a box in one of the sets, the second set may have several boxes in clusters near a single root. Thus, the output is of higher quality if there are relatively more boxes in the first set. In contrast to previously implemented similar techniques, a box expansion technique in this paper, based on using an approximate root finder, fflinflation and exact set complementation, decreases the size of the second set, increases the size of the first set, and never loses roots. In addition to the expansion technique, use of secondorder extensions to eliminate small boxes that do not contain roots, and interval slopes versus interval d...
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 16 (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 on 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 coecient matrix A = [A, A] is an Hmatr ..."
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Cited by 15 (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 coecient 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 6 = 0. Earlier results of others also imply that, when A is an Mmatrix and b 0,b 0, or 0 2 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.
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, "H ..."
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Cited by 13 (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...
Guaranteed robust nonlinear estimation, with application to robot localization
 IEEE Transactions on systems, man and cybernetics; Part C – Applications and Reviews 32 (4) (2003) 374—382, accepted
"... Abstract—When reliable prior bounds on the acceptable errors between the data and corresponding model outputs are available, boundederror estimation techniques make it possible to characterize the set of all acceptable parameter vectors in a guaranteed way, even when the model is nonlinear and the ..."
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Cited by 12 (4 self)
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Abstract—When reliable prior bounds on the acceptable errors between the data and corresponding model outputs are available, boundederror estimation techniques make it possible to characterize the set of all acceptable parameter vectors in a guaranteed way, even when the model is nonlinear and the number of data points small. However, when the data may contain outliers, i.e., data points for which these bounds should be violated, this set may turn out to be empty, or at least unrealistically small. The outlier minimal number estimator (OMNE) has been designed to deal with such a situation, by minimizing the number of data points considered as outliers. OMNE has been shown in previous papers to be remarkably robust, even to a majority of outliers. Up to now, it was implemented by random scanning, so its results could not be guaranteed. In this paper, a new algorithm based on set inversion via interval analysis provides a guaranteed OMNE, which is applied to the initial localization of an actual robot in a partially known twodimensional (2D) environment. The difficult problems of associating range data to landmarks of the environment and of detecting potential outliers are solved as byproducts of the procedure. I.
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 12 (9 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
On Proving Existence of Feasible Points in Equality Constrained Optimization Problems
 Mathematical Programming
, 1995
"... Various algorithms can compute approximate feasible points or approximate solutions to equality and bound constrained optimization problems. In exhaustive search algorithms for global optimizers and other contexts, it is of interest to construct bounds around such approximate feasible points, then t ..."
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Cited by 12 (6 self)
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Various algorithms can compute approximate feasible points or approximate solutions to equality and bound constrained optimization problems. In exhaustive search algorithms for global optimizers and other contexts, it is of interest to construct bounds around such approximate feasible points, then to verify (computationally but rigorously) that an actual feasible point exists within these bounds. Hansen and others have proposed techniques for proving the existence of feasible points within given bounds, but practical implementations have not, to our knowledge, previously been described. Various alternatives are possible in such an implementation, and details must be carefully considered. Also, in addition to Hansen's technique for handling the underdetermined case, it is important to handle the overdetermined case, when the approximate feasible point corresponds to a point with many active bound constraints. The basic ideas, along with experimental results from an actual implementation...
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 11 (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.