Quasiconvex relaxations based on interval arithmetic

Interval analysis provides a tool for (i) forward error analysis, (ii) esti mating and controlling rounding and approximation errors automatically, and (iii) proving existence and uniqueness of solutions. In this context the t erms self-validating methods, inclusion methods or verification methods are in use. In this paper, we present a new self-validating method for solving global constrained optimization problems. This method is based on the cons truction of quasiconvex lower bound and quasiconcave upper bound functions of a given function, the latter defined by an arithmetical expression. No f urther assumptions about the nonlinearities of the given function are neces sary. These lower and upper bound functions are rigorous by using the tools of interval arithmetic. In its easiest form they are constructed by taking appropriate linear and/or quadratical estimators which yield quasiconvex/q uasiconcave bound functions. We show how these bound functions can be used to define rigorous quasiconvex relaxations for constrained global optimizat ion problems and nonlinear systems. These relaxations can be incorporated i n a branch and bound framework yielding a self-validating method. (C) 2001 Elsevier Science Inc. All rights reserved. AMS classification: 90C26; 65G10 .