Efficient use of optimality conditions in Interval Branch and Bound methods

Abstract

The Interval Branch and Bound (IBB) method is a widely used approach for solving nonlinear programming problems where a rigorous solution is required. The method uses Interval Arithmetic (IA) to handle rounding errors in calculations. In the literature, a wide range of variations of IBB exists. However, few IBB implementations use the Karush-Kuhn-Tucker (KKT) or the Fritz-John (FJ) optimality conditions to eliminate non-optimal boxes. The application of the FJ conditions implies to solve a system of interval linear equations, which is often challenging due to overestimation of the boxes. This study focuses on the geometric perspective of the FJ optimality conditions. A preliminary test is introduced, namely the Geometrical Test, which tries to decide when the optimality conditions cannot hold or whether it is convenient to compute the Fritz-John Test. Furthermore, a test case generator is presented that transforms unconstrained problems into constrained test cases by setting a given number of active and inactive constraints at a global optimizer. The efficiency of the Geometrical Test was considered through computational experiments on the generated benchmark. Six variations of the IBB were compared, with or without the FJ condition system and Geometrical Test. The best methods for solving the 272 generated test cases use the designed Geometrical Test with the Lagrange estimator and the Newton step on the normalized interval FJ conditions in most cases.