Interval methods for non-linear constraints

Abstract

Many problems in areas as diverse as chemical engineering, economics, logistics, kinematics, statistics or even nuclear engineering are naturally expressed in terms of non-linear equations. Non-linear equations solving, or more generally global optimization, is a difficult problem because of its computational complexity and its numerical pitfalls. Global optimization has been studied for many years. The state of the art offers a wide variety of algorithmic solutions that encompass a number of goals.

The purpose of the tutorial is to introduce the latest techniques used for non-linear constraints. It covers continuation methods from numerical analysis, classical Newton methods and interval methods. It presents each of them and discusses the various strength and weaknesses. These methods are then combined with techniques inspired from artificial intelligence like box-consistency, an approximation of arc-consistency, and bound-consistency. This mix is stirred into a global method, the Interval Newton method. The combination has many advantages to offer. It brings a global component that ensures completeness while the intervals offer soundness. This entails an elegant semantics with strong guarantees on the computation results.

On the other hand, the numerical analysis stronghold plays a major role in the efficiency of the method. Many tools can be adapted and used to improve both the efficiency and the accuracy. The Taylor expansion is a remarkable example. It can be turned into an interval extension just like normal functions are. Moreover, its global nature introduces yet another pruning operator that blends quite nicely with the others. It prunes under different circumstances and helps in establishing the proofs of existence for solutions. Other ideas like the utilization of redundant information has often proved worthy. This track is followed again with the partial Groebner basis that introduces redundant polynomials.

Traditional local techniques can also find their way back under the hood, reinforced by the interval component. This last addition allows better optimization techniques and also gives the power to obtain a single local solution with a well-defined semantics.