The Concave-Convex Procedure (CCCP)

This paper describes a simple geometrical Concave-Convex procedure (CCCP) for constructing discrete time dynamical systems which can be guaranteed to decrease almost any global optimization/energy function (see technical conditions in section (2)). We prove that there is a relationship between CCCP and optimization techniques based on introducing auxiliary variables using Legendre transforms. We distinguish between Legendre min-max and Legendre minimization. In the former, see [6], the introduction of auxiliary variables converts the problem to a min-max problem where the goal is to find a saddle point. By contrast, in Legendre minimization, see [8], the problem remains a minimization one (and so it becomes easier to analyze convergence).