Interval Arithmetic Techniques Research Articles

Global Interval Methods for Local Nonsmooth Optimization

An interval method for determining local solutions of nonsmooth unconstrained optimization problems is discussed. The objective function is assumed to be locally Lipschitz and to have appropriate interval inclusions. The method consists of two parts, a local search and a global continuation and termination. The local search consists of a globally convergent descent algorithm showing similarities to e-bundle methods. While e-bundle methods use polytopes as inner approximations of the e-subdifferentials, which are the main tools of almost all bundle concepts, our method uses axes parallel boxes as outer approximations of the e-subdifferentials. The boxes are determined almost automatically with inclusion techniques of interval arithmetic. The dimension of the boxes is equal to the dimension of the problem and remains constant during the whole computation. The application of boxes does not suffer from the necessity to invest methodical and computational efforts to adapt the polytopes to the latest state of the computation as well as to simplify them when the number of vertices becomes too large, as is the case with the polytopes. The second part of the method applies interval techniques of global optimization to the approximative local solution obtained from the search of the first part in order to determine guaranteed error bounds or to improve the solution if necessary. We present prototype algorithms for both parts of the method as well as a complete convergence theory for them and demonstrate how outer approximations can be obtained.

Constraint reasoning based on interval arithmetic: the tolerance propagation approach

Interval constraint satisfaction (interval labeling) systems have traditionally been based on local Waltz filtering techniques that cannot in general determine global solutions. In contrast, this paper documents a related technique, tolerance propagation (TP), that generalizes the idea of numerical exact value propagation into interval propagation. In TP, consistency techniques based on the topology of the constraint net can be combined with techniques of interval arithmetic in a new fruitful way. In particular, by TP it is possible to determine global solutions for interval constraint satisfaction problems with arbitrary accuracy and without losing all attractions of simple local computations.

Precise Scientific Computation with a Microprocessor

When a high-level programming language is used for scientific computation, usually one must use standard floating point and choose the precision-single, double, or higher-before one's program executes. This paper describes a modified Basic that accommodates the advanced computation techniques of dynamic adjustable precision and interval arithmetic. This language-Precision Basic or Pbasic-allows a user to routinely obtain his n decimal place answers accurate to the last decimal place. An interpreter for this modified Basic was recently completed that runs on a Z-80 microprocessor. Some features of the interpreter are described.

Bounding Global Minima with Interval Arithmetic

We show how techniques of interval arithmetic can be used to give bounds on the global minima of unconstrained optimization problems. We illustrate the techniques using the design of a hypothetical chemical plant.