Abstract
In the paper the new formulation of the Broyden restricted convex class of updates involving oblique projections and some of its properties are presented. The new formulation involves two oblique projections. The new formula is a sum of two terms. The first one have the product form similar to that known for years for the famous BFGS (Broyden, Fletcher, Goldfarb, Shanno) update. The difference is that the oblique projection in the product contains vector defined as the convex, linear combination of the difference between consecutive iterative points and the image of the previous inverse hessian approximation on the corresponding difference of derivatives, i.e. gradients. The second standard term ensuring verification of the quasi-Newton condition is also an oblique projection multiplied by appropriate scalar. The formula relating the scalar parameter in the presented new version of updates with the formula appearing in the standard formulation is introduced and analyzed analytically and graphically. Formal proof of the theoretical equivalence of both updating formulas, when this relation is verified, is presented.
Some preliminary numerical experiments results on two twice continuously differentiable, strictly convex functions with increasing dimension are included.
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Stachurski, A. (2013). Some Properties of the Broyden Restricted Class of Updates with Oblique Projections. In: Fidanova, S. (eds) Recent Advances in Computational Optimization. Studies in Computational Intelligence, vol 470. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00410-5_10
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DOI: https://doi.org/10.1007/978-3-319-00410-5_10
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