The use of interval methods, in particular interval-Newton/generalized-bisection techniques, provides an approach that is mathematically and computationally guaranteed to reliably solve difficult nonlinear equation solving and global optimization problems, such as those that arise in chemical process modeling. The most significant drawback of the currently used interval methods is the potentially high computational cost that must be paid to obtain the mathematical and computational guarantees of certainty. New methodologies are described here for improving the efficiency of the interval approach. In particular, a new hybrid preconditioning strategy, in which a simple pivoting preconditioner is used in combination with the standard inverse-midpoint method, is presented, as is a new scheme for selection of the real point used in formulating the interval-Newton equation. These techniques can be implemented with relatively little computational overhead, and lead to a large reduction in the number of subintervals that must be tested during the intervalNewton procedure. Tests on a variety of problems arising in chemical process modeling have shown that the new methodologies lead to substantial reductions in computation time requirements, in many cases by multiple orders of magnitude.

A strategy is described for using linear programming (LP) to bound the solution set of the linear interval equation system that must be solved in the context of the interval-Newton method for deterministic global optimization. An implementation of this technique is described in detail, and several important issues are considered. These include selection of the interval corner required by the LP strategy, and determination of rigorous bounds on the solutions of the LP problems. The impact of using a local minimizer for updating the upper bound on the global minimum in this context is also considered. The procedure based on these techniques, LISS LP, is demonstrated using several global optimization problems, with focus on problems arising in chemical engineering. Problems with a very large number of local optima can be effectively solved, as well as problems with a relatively large number of variables.

. 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...

This paper is an introduction to Newton, a constraint programming language over nonlinear real constraints. Newton originates from an eort to reconcile the declarative nature of constraint logic programming (CLP) languages over intervals with advanced interval techniques developed in numerical analysis, such as the interval Newton method. Its key conceptual idea is to introduce the notion of box-consistency, which approximates arc-consistency, a notion well-known in articial intelligence. Box-consistency achieves an eective pruning at a reasonable computation cost and generalizes some traditional interval operators. Newton has been applied to numerous applications in science and engineering, including nonlinear equation-solving, unconstrained optimization, and constrained optimization. It is competitive with continuation methods on their equation-solving benchmarks and outperforms the interval-based methods we are aware of on optimization problems. Key words: Constraint Programming, Nonlinear Programming, Interval Reasoning 1 Introduction Many applications in science and engineering (e.g., chemistry, robotics, economics, mechanics) require nding all isolated solutions to a system of nonlinear real constraints or nding the minimum value of a nonlinear function subject to nonlinear constraints. These problems are dicult due to their inherent computational complexity (i.e., they are NP-hard) and due to the numerical issues involved to guarantee correctness (i.e., nding all solutions or the global optimum) and to ensure termination. Preprint submitted to Elsevier Preprint 11 June 2001 Newton is a constraint programming language designed to support this class of applications. It originates from an attempt to reconcile the declarative nature of CLP(Intervals) languag...

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...

In recent years, it has been shown that strategies based on an interval-Newton approach can be used to reliably solve a variety of nonlinear equation solving and optimization problems in chemical process engineering, including problems in parameter estimation and in the computation of phase behavior. These strategies provide a mathematical and computational guarantee either that all solutions have been located in an equation solving problem or that the global optimum has been found in an optimization problem. The primary drawback to this approach is the potentially high computational cost. In this paper, we consider strategies for bounding the solution set of the linear interval equation system that must be solved in the context of the interval-Newton method. Recent preconditioning techniques for this purpose are reviewed, and a new bounding approach based on the use of linear programming (LP) techniques is presented. Using this approach it is possible to determine the desired bounds exactly (within round out), leading to significant overall improvements in computational efficiency. These techniques will be demonstrated using several global optimization problems, with focus on problems arising in chemical engineering, including parameter estimation and molecular modeling. These problems range in size from under ten variables to over two hundred, and are solved deterministically using the interval methodology.

Techniques for verifying feasibility of equality constraints are presented. The underlying verification procedures are similar to a proposed algorithm of Hansen, but various possibilities, as well as additional procedures for handling bound constraints, are investigated. The overall scheme differs from some algorithms in that it rigorously verifies exact (rather than approximate) feasibility. The scheme starts with an approximate feasible point, then constructs a box (i.e. a set of tolerances) about this point within which it is rigorously verified that a feasible point exists. Alternate ways of proceeding are compared, and numerical results on a set of test problems appear.

A rigorous and efficient algorithm is presented for computing a sequence of points on all the branches of surface patch intersection curves within a given box. In the algorithm, an interval step control continuation method makes certain that the predictor algorithm will not jump from one branch to the another. These reliability properties are independent of any choice of tuning parameters. Both a 3-dimensional box complement method and a containment checking method are able to guarantee that all branches are located. Initial experimental results show that, even with this reliability, the amount of computation is orders of magnitude less than a uniform tesselation of the three-dimensional viewing box. Keywords: computational geometry, marching method, continuation method, surface patch intersections, interval computations. 1 Introduction and Notation The goal of this paper is to present general algorithms for computing all surface / surface intersection curves that are mathematically ...

We consider the problem of finding interval enclosures of all zeros of a nonlinear system of polynomial equations. We present a method which combines the method of Grobner bases (used as a preprocessing step), some techniques from interval analysis, and a special version of the algorithm of E. Hansen for solving nonlinear equations in one variable. The latter is applied to a triangular form of the system of equations, which is generated by the preprocessing step. Our method is able to check if the given system has a finite number of zeros and to compute verified enclosures for all these zeros. Several test results demonstrate that our method is much faster than the application of Hansen's multidimensional algorithm (or similar methods) to the original nonlinear systems of polynomial equations. 1 Introduction The general problem we address is: Find, with certainty, all solutions in IR n of the nonlinear system f k (x 1 ; x 2 ; : : : ; x n ) = 0 for k = 1; : : : ; m of m ...

An algorithm for computing inclusions for all global minimizers of a function f : IR n ! IR with f 2 C 2 (IR n ) in a compact interval vector and its implementation in PASCAL--XSC are presented. The algorithm is based on the method of E. Hansen using branch-and-bound techniques and interval arithmetic. First, some basic concepts, the essential parts of the algorithm, and some modifications are described. Subsequently, the comfortable programming of the algorithm in PASCAL--XSC and the easy use of the optimization program is demonstrated. Due to the C-based implementation of PASCAL--XSC the compiler as well as the optimization algorithm is portable. Thus, numerical results and performance tests of different computer types are presented for some optimization problems. 1 Introduction We present an implementation of an algorithm for computing guaranteed bounds for all solutions of the global unconstrained optimization problem minf(x) subject to x 2 X; (1) where X ` IR n is a com...