It is known that quasi-Newton updates can be characterized by variational means, sometimes in more than one way. This paper has two main goals. We first formulate variational problems appearing in quasi-Newton methods within the vector space of symmetric matrices. This simplifies both their formulations and their subsequent solutions. We then construct, for the first time, duals of the variational problems for the DFP and BFGS updates and discover that the solution to a dual problem is either the same as the corresponding primal solution or the solutions are inverses of each other. Consequently, we obtain six new variational characterizations for the DFP and BFGS updates, three for each one.
A convex body K in R n has around it a unique circumscribed ellipsoid CE(K) with minimum volume, and within it a unique inscribed ellipsoid IE(K) with maximum volume. The modern theory of these ellipsoids is pioneered by Fritz John in his seminal 1948 paper. This paper has two, related goals. First, we investigate the symmetry properties of a convex body by studying its (affine) automorphism group Aut(K), and relate this group to the automorphism groups of its ellipsoids. We show that if Aut(K) is large enough, then the complexity of determining the ellipsoids CE(K) and IE(K) is greatly reduced, and in some cases, the ellipsoids can be determined explicitly. We then use this technique to compute the extremal ellipsoids associated with some classes of convex bodies that have important applications in convex optimization, namely when the convex body K is the part of a given ellipsoid between two parallel hyperplanes, and when K is a truncated second-order cone or an ellipsoidal cylinder.
In this work, we develop duality of the minimum volume circumscribed ellipsoid and the maximum volume inscribed ellipsoid problems. We present a unified treatment of both problems using convex semi-infinite programming. We establish the known duality relationship between the minimum volume circumscribed ellipsoid problem and the optimal experimental design problem in statistics. The duality results are obtained using convex duality for semi-infinite programming developed in a functional analysis setting.
In this paper, an effective process optimization approach based on artificial neural networks with a back propagation algorithm and response surface methodology including central composite design is presented for the modeling and prediction of surface roughness in the wire electrical discharge machining process. In the development of predictive models, cutting parameters of pulse duration, open circuit voltage, wire speed and dielectric flushing are considered as model variables. After experiments are carried out, the analysis of variance is implemented to identify the contribution of uncontrollable process parameters effecting surface roughness. Then, a comparative analysis of the proposed approaches is carried out to determine the most efficient one. The performance of the developed artificial neural networks and response surface methodology predictive models is tested for prediction accuracy in terms of the coefficient of determination and root mean square error metrics. The results indicate that an artificial neural networks model provides more accurate prediction than the response surface methodology model.
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