Of the various interval-based software packages that are available, we chose INTLAB for several reasons. It is fully integrated into the interactive, programmable, and highly popular MATLAB system. It is carefully written, with all basic interval computations represented. Finally, both MATLAB and INTLAB code can be written in a fashion that is clear and easy to debug. We wish to cordially thank George Corliss, Andreas Frommer, and Siegfried Rump, as well as the anonymous reviewers, for their many constructive comments. We owe Siegfried Rump additional thanks for developing INTLAB and granting us permission to use it in this book.

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Interval Newton methods in conjunction with generalized bisection can form the basis of algorithms that find all real roots within a specified box X C p, on of a system of nonlinear equations F(X) 0 with mathematical certainty, even in finite-precision arithmetic. In such methods, the system F(X) 0 is transformed into a linear interval system 0 F(M) + FP(X)(X-M); if interval arithmetic is then used to bound the solutions of this system, the resulting box X contains all roots of the nonlinear system. The interval Gauss-Seidel method is a reasonable way of finding such solution bounds.For the overall interval Newton/bisection algorithm to be efficient, the image box X should be as small as possible. To do this, the linear interval system is multiplied by a preconditioner matrix Y before the interval Gauss-Seidel method is applied. In this paper, a technique for computing such preconditioner matrices Y is described. This technique involves optimality conditions expressible as linear programming problems. In many instances, the resulting preconditioners give an X of minimal width. They can also be applied when F approximates a singular matrix, and the optimality conditions can be altered to describe preconditioners with a given structure. This technique is illustrated with some simple examples and with numerical experiments. These experiments indicate that the new preconditioner results in significantly less function and Jacobian evaluations, especially for ill-conditioned problems, but it requires more computation to obtain.

We present a portable software package for finding all real roots of a system of nonlinear equations within a region defined by bounds on the variables. Where practical, the package should find all roots with mathematical certainty. Though based on interval Newton methods, it is self-contained. It allows various control and output options and does not require programming if the equations are polynomials; it is structured for further algorithmic research. Its practicality does not depend in a simple way on the dimension of the system or on the degree of nonlinearity.

We consider branch and bound methods for enclosing all unconstrained global minimizers of a nonconvex nonlinear twice-continuously differentiable objective function. In particular, we consider bounds obtained with interval arithmetic, with the "midpoint test," but no acceleration procedures. Unless the lower bound is exact, the algorithm without acceleration procedures in general gives an undesirable cluster of boxes around each minimizer. In a previous paper, we analyzed this problem for univariate objective functions. In this paper, we generalize that analysis to multidimensional objective functions. As in the univariate case, the results show that the problem is highly related to the behavior of the objective function near the global minimizers and to the order of the corresponding interval extension. We will refer to X as a box. We denote the global minimum as f * and the set of global minimizers as X *. Interval branch and bound procedures for unconstrained nonconvex optimization, i.e. for rigorously enclosing the solution set X * of (1), are competitive with stochastic methods, such as Monte Carlo methods, and methods involving heuristics, such as simulated annealing or the tunneling method. Our analysis deals with algorithms similar to Algorithm 3, p. 111 of [10]. Also, as in [10], we will use interval arithmetic to obtain the bounds. This paper deals with the phenomenon of clusters of small boxes around global minimizers that such algorithms produce cannot eliminate. We refer to this phenomenon as the cluster problem. Though the algorithms in which the cluster problem occurs apply to global, nonconvex, unconstrained optimization, the cluster problem is essentially a local phenomenon dealing with the portion of the domain Key words and phrases. branch and bound principle, inclusion function, interval extensions, midpoint test, global optimization, order of an interval extension, nonconvex optimization.