The Interval Branch and Bound (IBB) method is a widely used approach for solving nonlinear programming problems, especially when a rigorous solution is required. It uses Interval Arithmetic to handle rounding errors. Although numerous variants of the IBB method have been proposed in the literature, relatively few implementations incorporate Karush-Kuhn-Tucker or Fritz-John (FJ) optimality conditions to eliminate nonoptimal boxes. Using the FJ conditions involves solving an interval-valued linear system of equations. However, the solution is often overestimated, making the test ineffective.
This study is based on the geometrical interpretation of the optimality condition and involves the development of an Advanced Geometrical Test to be performed before solving the optimality conditions. The theoretical validity of the new techniques has been established, and the efficiency of the new test was evaluated through computational experiments on a benchmark set of 374 test cases. Nine variants of the IBB algorithm were compared, differing only in the tests used to check the optimality conditions. The IBB method combined with the Advanced Geometrical Test and the Lagrange estimator method achieves the best overall performance, earning a 2.1 times speed-up.