A modification of the αBB method for box-constrained optimization and an application to inverse kinematics

Abstract

For many practical applications it is important to determine not only a numerical approximation of one but a representation of the whole set of globally optimal solutions of a non-convex optimization problem. Then one element of this representation may be chosen based on additional information which cannot be formulated as a mathematical function or within a hierarchical problem formulation. We present such an application in the field of robotic design. This application problem can be modeled as a smooth box-constrained optimization problem. We extend the well-known $$\alpha \hbox {BB}$$ method such that it can be used to find an approximation of the set of globally optimal solutions with a predefined quality. We illustrate the properties and give a proof for the finiteness and correctness of our modified $$\alpha \hbox {BB}$$ method.