Circuit optimization via adjoint Lagrangians

The circuit tuning problem is best approached by means of gradient-based nonlinear optimization algorithms. For large circuits, gradient computation can be the bottleneck in the optimization procedure. Traditionally, when the number of measurements is large relative to the number of tunable parameters, the direct method is used to repeatedly solve the associated sensitivity circuit to obtain all the necessary gradients. Likewise, when the parameters outnumber the measurements, the adjoint method is employed to solve the adjoint circuit repeatedly for each measurement to compute the sensitivities. In this paper, we propose the adjoint Lagrangian method, which computes all the gradients necessary for augmented-Lagrangian-based optimization in a single adjoint analysis. After the nominal simulation of the circuit has been carried out, the gradients of the merit function are expressed as the gradients of a weighted sum of circuit measurements. The weights are dependent on the nominal solution and on optimizer quantities such as Lagrange multipliers. By suitably choosing the excitations of the adjoint circuit, the gradients of the merit function are computed via a single adjoint analysis, irrespective of the number of measurements and the number of parameters of the optimization. This procedure requires close integration between the nonlinear optimization software and the circuit simulation program. The adjoint Lagrangian formulation has been implemented in the JiffyTune tool which optimizes delay, area, slew (transition time) and power measurements by adjusting transistor widths and wire sizes. Speedups of over 35x have been realized in the gradient computation procedure by using the adjoint Lagrangian formulation, leading to speedups of up to 2.5x in the overall optimization procedure. Perhaps more importantly, these speedups have rendered feasible the tuning of large circuits. A circuit with 6,900 transistors was optimized in under two hours of CPU time.