Nonsmooth Optimization Method for VLSI Global Placement

Abstract

The common objective of very large-scale integration (VLSI) placement problem is to minimize the total wirelength, which is calculated by the total half-perimeter wirelength (HPWL). Since the HPWL is not differentiable, various differentiable wirelength approximation functions have been proposed in analytical placement methods. In this paper, we reformulate the HPWL as an l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm model of the wirelength function, which is exact but nonsmooth. Based on the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm wirelength model and exact calculation of overlapping areas between cells and bins, a nonsmooth optimization model is proposed for the VLSI global placement problem, and a subgradient method is proposed for solving the nonsmooth optimization problem. Moreover, local convergence of the subgradient method is proved under some suitable conditions. In addition, two enhanced techniques, i.e., an adaptive parameter to control the step size and a cautious strategy for increasing the penalty parameter, are also used in the nonsmooth optimization method. In order to make the placement method scalable, a multilevel framework is adopted. In the clustering stage, the best choice clustering algorithm is modified according to the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm wirelength model to cluster the cells, and the nonsmooth optimization method is recursively used in the declustering stage. Comparisons of experimental results on the International Symposium on Physical Design (ISPD) 2005 and 2006 benchmarks show that the global placement method is promising.