Interval Tests and Contractors Based on Optimality Conditions for Bound-Constrained Global Optimization

We study the problem of finding the global optimum of a nonlinear real function over an interval box by means of complete search techniques, namely interval branch-and-bound algorithms. Such an algorithm typically generates a tree of boxes from the initial box by alternating branching steps and contraction steps in order to remove non optimal sub-boxes. In this paper, we introduce a new contraction method that is designed to handle the boundary of the initial box where a minimizer may not be a stationary point. This method exploits the first-order optimality conditions and we show that it subsumes the classical monotonicity test based on interval arithmetic. A new branch-and-bound algorithm has been implemented in the interval solver Realpaver. An extensive experimental study based on a set of standard benchmarks is presented.

Современные интервальные подходы к вычислению решений непрерывных игр

Computing Nash equilibria in continuous games is a difficult problem, but interval methods have already been applied to its solution quite successfully. The purpose of this paper is to briefly survey previous efforts and achievements of the author related to the topic, and to consider some advanced tools for accelerating the interval branch-and-bound-type methods. In particular, we discuss computing eigenvalues of interval matrices, use of algorithmic (automatic) differentiation, memory management techniques as well as advanced parallelization in both shared-memory and distributedmemory environments. Дан краткий анализ результатов исследований по применению интервальных методов для вычисления равновесия по Нэшу в непрерывных играх. Рассмотрены некоторые современные подходы к ускорению интервальных методов типа ветвей и границ. В частности, обсуждаются такие вопросы, как вычисление собственных значений интервальных матриц, использование алгоритмического(автоматического) дифференцирования, методы управления памятью, инструменты распараллеливания в средах с разделяемой и распределенной памятью.