CONVEXITY-BASED ALGORITHMS FOR DESIGN CENTERING

A new technique for design centering and polytope approximation of the feasible region for a design are presented. In the first phase, the f easible region is approximated by a convex polytope, using a method ba sed on a theorem on convex sets. As a natural consequence of this appr oach, a good approximation to the design center is obtained. In the ne xt phase, the exact design center is estimated using one of two techni ques that we present in this paper. The first inscribes the largest He ssian ellipsoid, which is known to be a good approximation to the shap e of the polytope, within the polytope. This represents an improvement over previous methods, such as simplicial approximation, where a hype rsphere or a crudely estimated ellipsoid is inscribed within the appro ximating polytope. However, when the probability density functions of the design parameters are known, the design center does not necessaily correspond to the center of the largest inscribed ellipsoid. Hence, a second technique is developed that incorporates the probability distr ibutions of the parameters, under the assumption that their variation is modeled by Gaussian probability distributions. The problem is formu lated as a convex programming problem and an efficient algorithm is us ed to calculate the design center, using fast and efficient Monte Carl o methods to estimate the yield gradient. An example is provided to il lustrate how ellipsoid-based methods fail to incorporate the probabili ty density functions and is solved using the convex programming-based algorithm.