An interval method for global inequality-constraint optimization problems

A simple interval method is suggested for globally solving optimization problems of the following type: minimize the objective function φo(x) subject to the functional constraints φo(x) ≤ 0, i=1,...r and the boundary constraints x∈ X(o) where x is a n-dimensional vector and X(o) is a given initial search region (a box). All the functions involved are assumed continuously differentiable in X(o). Known interval methods for solving the global optimization problem considered are iterative and are based on a specific interval linearization of all the functions associated with the problem at each iteration of the method. This linearization involves one real term and n interval terms. In contrast, the present method appeals to a new interval linearization of each function which is constructed in such a way that now only one additive term is an interval while the remaining n terms are real numbers. The algorithm of the iterative method suggested includes the following basic procedures to be carried out at each iteration: (i) if the current subbox X is strictly feasible, then several tests for unconstrained minimization are applied and an upper bound on the global minimum is also found; (ii) if X is infeasible, X is discarded; (iii) if the first two cases do not occur, then a system of inequalities, including the objective function and some of the functional inequalities, is linearized and simultaneously solved in an attempt to reduce or completely discard X. The system involves at most n inequalities. The use of the new interval linearization in the computational scheme of the present method leads to improved performance as compared to other known interval methods of the same class. CSCC'99 Proc.pp.3661-3664 Key-Words: global optimization, inequality-constraint problems, interval methods, interval linearization.