PolyGP: Improving GP-Based Analog Optimization through Accurate High-Order Monomials and Semidefinite Relaxation

Geometric programming (GP) is popular for use in equation-based optimization of analog circuits thanks to GP-compatible analog performance functions, and its convexity, hence computational tractability. The main challenge in using GP, and thus a roadblock to wider use and adoption, is the mismatch between what GP can accurately fit, and the behavior of many common device/circuit functions. In this paper, we leverage recent tools from sums-of-squares, moment optimization, and semidefinite optimization (SDP), in order to present a novel and powerful extension to address the monomial inaccuracy: fitting device models as higher-order monomials, defined as the exponential functions of polynomials in the logarithmic variables. By the introduction of high-order monomials, the original GP problems become polynomial geometric programming (PolyGP) problems with non-linear and non-convex objective and constraints. Our PolyGP framework allows significant improvements in model accuracy when symbolic performance functions in terms of device models are present. Via SDP-relaxations inspired by polynomial optimization (POP), we can obtain efficient near-optimal global solutions to the resulting PolyGP. Experimental results through established circuits show that compared to GP, we are able to reduce fitting error of device models to 3.5% from 10.5% on average. Hence, the fitting error of performance functions decrease from 12% of GP and 9% of POP, to 3% accordingly. This translates to the ability of identifying superior solution points and the dramatic decrease of constraint violation in contrast to both GP and POP.