ConvexSmooth: A simultaneous convex fitting and smoothing algorithm for convex optimization problems

Convex optimization problems are very popular in the VLSI design society due to their guaranteed convergence to a global optimal point. Table data is often fitted into analytical forms like posynomials to make them convex. However, fitting the look-up tables into posynomial forms with minimum error itself may not be a convex optimization problem and hence excessive fitting errors may be introduced. In recent literature numerically convex tables have been proposed. These tables are created optimally by minimizing the perturbation of data to make them numerically convex. But since these tables are numerical, it is extremely important to make the table data smooth, and yet preserve its convexity. Smoothness will ensure that the convex optimizer behaves in a predictable way and converges quickly to the global optimal point. In this paper, we propose to simultaneously create optimal numerically convex look-up tables and guarantee smoothness in the data. We show that numerically "convexifying" and "smoothing" the table data with minimum perturbation can be formulated as a convex semidefinite optimization problem and hence optimality can be reached in polynomial time. We present our convexifying and smoothing results on industrial cell libraries. ConvexSmooth shows 14X reduction in fitting error over a well-developed posynomial fitting algorithm.