Quasiconvex relaxations based on interval arithmetic

Abstract

Interval analysis provides a tool for (i) forward error analysis, (ii) estimating and controlling rounding and approximation errors automatically, and (iii) proving existence and uniqueness of solutions. In this context the terms self-validating methods, inclusion methods orverification 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 construction of quasiconvex lower bound and quasiconcave upper bound functions of a given function, the latter defined by an arithmetical expression. No further assumptions about the nonlinearities of the given function are necessary. 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/quasiconcave bound functions. We show how these bound functions can be used to define rigorous quasiconvex relaxations for constrained global optimization problems and nonlinear systems. These relaxations can be incorporated in a branch and bound framework yielding a self-validating method.