In this paper, the deterministic global optimization algorithm, αBB, (α-based 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 twice-differentiable 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...

. 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...

. Inmum-supremum interval arithmetic is widely used because of ease of implementation and narrow results. In this note we show that the overestimation of midpoint-radius 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 midpoint-radius 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, midpoint-radius, inmum-supremum, AMS subject classications. 65G10 1. Introduction and notati...

This paper addresses the construction of relaxations for problems involving multivariate polynomials. The major goal is to show how non-convex 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...

Part I of this paper (Adjiman et al., 1997) described the theoretical foundations of a global optimization algorithm, the ffBB algorithm, which can be used to solve problems belonging to the broad class of twice-differentiable NPLs. For any such problem, the ability to automatically generate progressively tighter convex lower bounding problems at each iteration guarantees the convergence of the branchand -bound ffBB algorithm to within ffl of the global optimum solution. Several methods were presented for the construction of convex valid underestimators for general nonconvex functions. In this second part, the performance of the proposed algorithm and its alternative underestimators is studied through their application to a variety of problems. An implementation of the ffBB is described and a number of rules for branching variable selection and variable bound updates are shown to enhance convergence rates. A user-friendly parser facilitates problem input and provides flexibility in the...

this documentation. 4 The class interval

ABSTRACT. This paper derives inequalities for general linear recurrences. Optimal bounds for solutions to the recurrence are obtained when the coefficients of the recursion lie in intervals that include zero. An important aspect of the derived bounds is that they are easily computable. The results bound solutions of triangular matrix equations and coefficients of ratios of power series.

A global optimization algorithm, αBB, for twice-differentiable NLPs is presented. It operates within a branch-and-bound framework and requires the construction of a convex lower bounding problem. A technique to generate such a valid convex underestimator for arbitrary twice-differentiable functions is described. The αBB has been applied to a variety of problems and a summary of the results obtained is provided.

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 ...

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 ill-conditioned. 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 ill-posed linear programming problem.