An integer programming approach to optimal derivative accumulation

Abstract

In automatic differentiation, vertex elimination is one of the many methods for Jacobian accumulation and in general it can be much more efficient than the forward mode or reverse mode (Forth et al. ACM Trans Math Softw 30(3):266-299, 2004; Griewank and Walther, Evaluating derivatives: principles and techniques of algorithmic differentiation, SIAM, Philadelphia, 2008). However, finding the optimal vertex elimination sequence of a computational graph is a hard combinatorial optimization problem. In this paper, we propose to tackle this problem with an integer programming (IP) technique, and we develop an IP formulation for it. This enables us to use a standard integer optimization solver to find an optimal vertex elimination strategy. In addition, we have developed several bound-tightening and symmetry-breaking constraints to strengthen the basic IP formulation. We demonstrate the effectiveness of these enhancements through computational experiments. © 2012 Springer-Verlag.