ICNAAM-2004 Extended Abstracts 1 – 4 An Interval Branch-and-Prune Algorithm for Discrete Minimax Problems D.G. Sotiropoulos ∗ and T.N. Grapsa University of Patras, Department of Mathematics, GR-265 04 Rio, Patras, Greece. We present an interval branch-and-prune method for a discrete minimax problem where the constituent minimax functions are continuously differentiable functions of one real variable. Our approach is based on smoothing the max-type function by exploiting the Jaynes’s maximum entropy [Phys. Rev., 106:620–630, 1957]. Our algorithm works within the branch and bound framework and incorporates a new accelerating device that composes inner and outer pruning steps. Pruning is achieved by using first order information of the entropic objective function by means of an interval evaluation. Our algorithm was implemented and tested on a complete test set and numerical results are presented. 1 Introduction There are many applications in engineering where a not necessarily differentiable objective function has to be optimized. The discrete minimax problem is such an example. The purpose of this paper is to describe a reliable method for solving the minimax optimization problem min x∈X max 1≤i≤m f i (x), (1) where f i : D⊆ R → R (i =1,...,m) are continuously differentiable functions, D is the closure of a nonempty bounded open subset of R, and X ⊆D is a search interval representing bound constraints for x. The aim is to find the global minimum f ∗ and the set X = {x ∗ ∈ X : f (x ∗ )= f ∗ } of all global minimizers of the objective function f (x) = max{f 1 (x),...,f m (x)}. Interval methods for global optimization combine interval arithmetic with the so-called branch- and-bound principle. These methods subdivide the search region in subregions (branches) and use bounds for the objective function to exclude from consideration subregions where a global min- imizer cannot lie. Interval arithmetic provides the possibility to compute such rigorous bounds automatically. Moreover, when the objective function is continuously differentiable, interval en- closures of the derivative along with a mean value form can be used to improve the enclosure of the function range. In a recent paper an interval branch-and-prune algorithm for computing verified enclosures for the global minimum and all global minimizers has been developed for the one-dimensional case [2]. In relation to this previous work, the present study is specialized to the minimax prob- lem (1). Our approach is based on smoothing the max-type function by exploiting the Jaynes’s maximum entropy [1] for transforming problem (1) into the problem of minimizing a continuously differentiable function. ∗ Corresponding author. E-mail: dgs@math.upatras.gr, Phone: +30 2610 997332, Fax: +30 2610 992965. c  2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim