An Improved Interval Linearization for Solving Nonlinear Problems

Let f(x) denote a system of n nonlinear functions in m variables, m≥n. Recently, a linearization of f(x) in a box x has been suggested in the form L(x)=Ax+b where A is a real n×m matrix and b is an interval n-dimensional vector. Here, an improved linearization L(x,y)=Ax+By+b, x∈x, y∈y is proposed where y is a p-dimensional vector belonging to the interval vector y while A and B are real matrices of appropriate dimensions and b is a real vector. The new linearization can be employed in solving various nonlinear problems: global solution of nonlinear systems, bounding the solution set of underdetermined systems of equations or systems of equalities and inequalities, global optimization. Numerical examples illustrating the superiority of L(x,y)=Ax+By+b over L(x)=Ax+b have been solved for the case where the problem is the global solution of a system of nonlinear equations (n=m).