An environment for general research into and prototyping of algorithms for reliable constrained and unconstrained global nonlinear optimization and reliable enclosure of all roots of nonlinear systems of equations, with or without inequality constraints, is being developed. This environment should be portable, easy to learn, use, and maintain, and sufficiently fast for some production work. The motivation, design principles, uses, and capabilities for this environment are outlined. The environment includes an interval data type, a symbolic form of automatic differentiation to obtain an internal representation for functions, a special technique to allow conditional branches with operator overloading and interval computations, and generic routines to give interval and non-interval function and derivative information. Some of these generic routines use a special version of the backward mode of automatic differentiation. The package also includes dynamic data structures for exhaustive sear...

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...

A branch and bound algorithm for computing globally optimal solutions to nonconvex nonlinear programs in continuous variables is presented. The algorithm is directly suitable for a wide class of problems arising in chemical engineering design. It can solve problems defined using algebraic functions and twice differentiable transcendental functions, in which finite upper and lower bounds can be placed on each variable. The algorithm uses rectangular partitions of the variable domain and a new bounding program based on convex/concave envelopes and positive definite combinations of quadratic terms. The algorithm is deterministic and obtains convergence with final regions of finite size. The partitioning strategy uses a sensitivity analysis of the bounding program to predict the best variable to split and the split location. Two versions of the algorithm are considered, the first using a local NLP algorithm (MINOS) and the second using a sequence of lower bounding programs in the search fo...

. 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...

Abstract. The GlobSol software package combines various ideas from interval analysis, automatic differentiation, and constraint propagation to provide verified solutions to unconstrained and constrained global optimization problems. After briefly reviewing some of these techniques and GlobSol’s development history, we provide the first overall description of GlobSol’s algorithm. Giving advice on use, we point out strengths and weaknesses in GlobSol’s approaches. Through examples, we show how to configure and use GlobSol.

A#ne arithmetic (AA) is a model for self-validated computation which, like standard interval arithmetic (IA), produces guaranteed enclosures for computed quantities, taking into account any uncertainties in the input data as well as all internal truncation and roundo# errors. Unlike standard IA, the quantity representations used by AA are first-order approximations, whose error is generally quadratic in the width of input intervals. In many practical applications, the higher asymptotic accuracy of AA more than compensates for the increased cost of its operations.

An algorithm for nding a large feasible n-dimensional interval for constrained global optimization is presented. The n-dimensional interval is iteratively enlarged about a seed point while maintaining feasibility. An interval subdivision method may be used to check feasibility of the growing box. The resultant feasible interval is constrained to lie within a given level set, thus ensuring it is close to the optimum. The ability to determine such a feasible interval is useful for exploring the neighbourhood of the optimum, and can be practically used in manufacturing considerations. The numerical properties of the algorithm are tested and demonstrated by an example problem.

This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 1984 by Eckmann, Koch and Wittwer, and independently studied and popularized since 1996 by Berz, Makino and Hoefkens. A highlight is their application to the verified integration of asteroid dynamics in the solar system in 2001, although the details given are not sufficient to check the validity of their claims.

Global optimization is a very hard problem especially when the number of variables is large (greater than several hundred). Recently, some methods including simulated annealing, branch and bound, and an interval Newton's method have made it possible to solve global optimization problems with several hundred variables. However, this is a small number of variables when one considers that integer programming can tackle problems with thousands of variables, and linear programming is able to solve problems with millions of variables. The goal of this research is to examine the present state of the art for algorithms to solve the unconstrained global optimization problem (GOP) and then to suggest some new approaches that allow problems of a larger size to be solved with an equivalent amount of computer time. This algorithm is then implemented using portable C++ and the software will be released for general use. This new algorithm is given with some theoretical results under which the algorit...

We show that faster solutions to unconstrained global optimization problems can be obtained by combining previous accelerations techniques for interval branch-andbound methods with affine arithmetic, a recent alternative to interval arithmetic that often provides tighter estimates. We support this claim by solving a few well-known problems.