Accelerating Global Optimization Methods: Exploring Local Convexity, Monotonicity and Uniqueness

Global optimization plays a key role in many areas but can be challenging due to the complexity of the objective function, the potential for multiple best solutions, and problem constraints. The focus here will be on continuous problems and deterministic methods such as branch and bound. This method explores the solution space by dividing it into smaller subboxes. However, as the method converges to the optimal solution, it produces more subboxes, leading to the cluster problem which increases the running time.
In this work, a new technique to avoid the cluster problem is introduced by combining a local iterative method with a global deterministic method. It is shown that this method can be used for nonlinear systems of equations and global optimization problems. Both theoretical analysis and numerical applications are provided, showing that the new combined method reduces the number of boxes to be examined and the computational time compared to the branch and bound method.