On the hypercomplex-based search spaces for optimization purposes

Abstract

Most applications can be modeled using real-valued algebra. Nevertheless, certain problems may be better addressed using different mathematical tools. In this context, complex numbers can be viewed as an alternative to standard algebra, where imaginary numbers allow a broader collection of tools to deal with different types of problems. In addition, hypercomplex numbers extend naïve complex algebra by means of additional imaginary numbers, such as quaternions and octonions. In this work, we will review the literature concerning hypercomplex spaces with an emphasis on the main concepts and fundamentals that build the quaternion and octonion algebra, and why they are interesting approaches that can overcome some potential drawbacks of certain optimization techniques. We show that quaternion- and octonion-based algebra can be used to different optimization problems, allowing smoother fitness landscapes and providing better results than those represented in standard search spaces.