We survey contemporary optimization techniques and relate these to optimization problems which arise in the design of integrated circuits. Theory, algorithms and programs are reviewed, and an assessment is made of the impact optimization has had and will have on integrated-circuit design. Integrated circuits are characterized by complex tradeoffs between multiple nonlinear objectives with multiple nonlinear and sometimes nonconvex constraints. Function and gradient evaluations require the solution of very large sets of nonlinear differential equations, consequently they are inaccurate and extremely expensive. Furthermore, the partmeters to be optimized are subject to inherent statistical fluctuations. We focus on those multiobjective constrained optimization techniques which are appropriate to this environment.

A survey of optimization techniques for integrated-circuit design

The authors review the current state of the art in circuit optimization, emphasizing techniques suitable for modern microwave CAD (computer-aided design). The main thrust in the field is currently the solution of realistic design and modeling problems, addressing such concepts as physical tolerances and model uncertainties. A unified hierarchical treatment of circuit models forms the basis of the presentation. It exposes tolerance phenomena at different parameter/response levels. The concepts of design centering, tolerance assignment, and postproduction tuning in relation to yield enhancement and cost reduction suitable for integrated circuits are discussed. Suitable techniques for optimization oriented worst-case and statistical design are reviewed. A generalized l/sub p/ centering algorithm is proposed and discussed. Multicircuit optimization directed at both CAD and robust device modeling is formalized. Tuning is addressed in some detail, both at the design stage and for production alignment. State-of-the-art gradient-based nonlinear optimization methods are reviewed with emphasis given to recent, but well tested, advances in minimax, l/sub 1/, and l/sub 2/ optimization. >

/pdf/circuit-optimization-the-state-of-the-art-2yre9izylm.pdf

Circuit optimization: the state of the art

The basis of a method for designing circuits in the face of parameter uncertainties is described. This method is computationally cheaper than those methods which employ Monte Carlo analysis and nonlinear programming techniques, gives more useful information, and more directly addresses the central problem of design centering. The method, called simplicial approximation, locates and approximates the boundary of the feasible region of an n -dimensional design space with a polyhedron of bounding ( n - 1 )-simplices. The design centering problem is solved by determining the location of the center of the maximal hyperellipsoid inscribed within this polyhedron. The axis lengths of this ellipsoid can be used to solve the tolerance assignment problem. In addition, this approximation can be used to estimate the yield by performing an inexpensive Monte Carlo analysis in the parameter space without any need for the usual multitude of circuit simulations.

The simplicial approximation approach to design centering

A new statistical circuit design centering and tolerancing methodology based on a synthesis of concepts from network analysis, recent optimization methods, sampling theory, and statistical estimation and hypothesis testing is presented. The method permits incorporation of such realistic manufacturing constraints as tuning, correlation, and end-of-life performance specifications. Changes in design specifications and component cost models can be handled with minimal additional computational requirements. A database containing the results of a few hundred network analyses is first constructed. As the nominal values and tolerances are changed by the optimizer, each new yield and its gradient are evaluated by a new method called Parametric sampling without resorting to additional network analyses. Thus the most costly phase of statistical design,-statistical simulation, may be carried out only once, which leads to considerable computational efficiency. Equivalent or superior designs for intermediate size networks are obtained with less computational effort than previously published methods. For example, a worst-case design for an eleventh-order Chebychev filter gives a filter cost of 44 units, a centered worst-case design reduces the cost to 18 units and statistical design using Parametric sampling further reduces the cost to 5 units (800 analyses, 75 CPU seconds on an IBM 370/158).

Statistical design centering and tolerancing using parametric sampling

The efficiency of several variance reduction techniques (in particular, importance sampling, stratified sampling, and control variates) are studied with respect to their application in estimating circuit yields. This study suggests that one essentially has to have a good approximation of the region of acceptability in order to achieve significant variance reduction. Further, all the methods considered are based, either explicitly or implicity, on the use of a model. The control variate method appears to be more practical for implementation in a general purpose statistical circuit analysis program. Stratified sampling is the most simple to implement, but yields only very modest reductions in the variance of the yield estimator. Lastly, importance sampling is very useful when there are few parameters and the yield is very high or very low; however, a good practical technique for its implementation, in general, has not been found.

A Study of Variance Reduction Techniques for Estimating Circuit Yields

Tolerance assignment in linear networks on a worstcase basis is treated as a nonlinear programming problem. The sequential unconstrained minimization technique of Fiacco and McCormick is employed as a solution mechanism with network behavior being evaluated by means of a large-change sensitivity model. The solution mechanism also yields worst-case responses at the constraint frequencies. Examples are presented and a comparison is made with the method proposed by Karafin.

K. Roberts

Papers

Tolerance assignment in linear networks using nonlinear programming