Pure Categorical Optimization - a Global Descent Approach

In this article we introduce and study the pure categorical optimization problem. This is a problem in discrete variables, but where the discrete variables have no natural ordering in the decision space. It is argued that such a problem is a natural framework for the study of, e.g., the problem of finding the right configuration for a customer for certain types of platform-based products. It is shown that the problem can be reformulated into several different nonlinear integer programming models that all are equivalent from a categorical point of view. We investigate its mathematical properties; in particular we establish properties corresponding to those of continuity and convexity for numerical optimization problems. For the solution of the problems we propose to extend the discrete global descent method from the area of (numerical) nonlinear integer programming. We suggest extensions of the principal algorithmic steps within these methods in order to adapt it to categorical problems, utilizing the fact that there are many equivalent problem formulations. Numerical results are provided when applying the proposed methods both to some standard test problems from the literature, and to real-world problems concerning the configuration of heavy-duty trucks. It is concluded that problems in this class can be solved using mathematical techniques. In particular, it is possible to utilize the discrete global descent method, and it is concluded that the extensions added in order to adapt it to categorical problems also improve its performance.