. The role of the interval subdivision selection rule is investigated in branch-and-bound 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...

Elements and techniques of state-of-the-art 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...

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 VI-th SERC Numerical Analysis Summer School, Leicester University, apply the principles of validated computatio...

We have investigated variants of interval branch-and-bound 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.

This paper presents some tools based on interval analysis for guaranteed nonlinear parameter and state estimation in a bounded-error 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.

. 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...

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...

INTLIB is meant to be a readily available, portable, exhaustively documented interval arithmetic library, written in standard FORTRAN 77. Its underlying philosophy is to provide a standard for interval operations to aid in efficiently transporting programs involving interval arithmetic. The model is the BLAS package, for basic linear algebra operations. The library is composed of elementary interval arithmetic routines, standard function routines for interval data and values, and utility routines. The library can be used with INTBIS (Algorithm 681), and a Fortran 90 module to use the library to define an interval data type is available from the first author. Keywords: interval arithmetic, standard functions, BLAS, operator overloading, FORTRAN 77, Fortran 90 Subject classifications: AMS: 65G10, 65D15, 26A09. CR: G.1.0 (Computer arithmetic), G.1.2 (standard function approximation), D.2.2 (Software libraries) D.2.7 (documentation, portability) This work is partially supported by Nat...

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 H-matrix; 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 M-matrix 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, inverse-positive matrix, H-matrix, interval hull, bounding solution sets

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 n-space. 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 rank-defect 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.