How to Guarantee Finite Termination of Verifying Global Optimization Codes

Abstract

Convergence or finite termination of algorithms for solving problems of numerical analysis usually is shown only for their original theoretical versions, assuming that all operations appearing in the algorithms can be performed exactly.

But in most cases the theoretical version of an algorithm cannot be realized on a computer and it is one of the main tasks of numerical mathematicians to find codeable versions of algorithms sharing the essential properties with their originals.

The main problem is, that in any coded version at most rounded approximations of the operations appearing in the theoretical algorithm can be used, and that these approximations are defined only for the finite set of floating point numbers.

So e.g. in codes for global optimization only rounded bisection is possible, and even this only finitely often in a meaningful way. In addition there are also problems with controlling rounding errors if underflow or graduated underflow might appear. These problems are often overlooked, even in codes for verifying global optimization.

But it is possible to solve these problems by a simple but quite general implementable procedure, guaranteeing that a very natural stopping criterion is satisfied after finitely many steps.