Some Properties of the Broyden Restricted Class of Updates with Oblique Projections

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.

[1]  Z. Nowak,et al.  Global optimization in material functions identification for voided media plastic flow , 2002 .

[2]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[3]  D. Shanno.  Conditioning of Quasi-Newton Methods for Function Minimization , 1970 .

[4]  Modelling and identification of voids nucleation and growth effects in porous media plastic flow , 2003 .

[5]  Andrzej STACHURSKI,et al.  Superlinear convergence of Broyden's boundedθ-class of methods , 1981, Math. Program..

[6]  J. Nocedal,et al.  Global Convergence of a Class of Quasi-newton Methods on Convex Problems, Siam Some Global Convergence Properties of a Variable Metric Algorithm for Minimization without Exact Line Searches, Nonlinear Programming, Edited , 1996 .

[7]  Zhiguo Wang,et al.  A limited memory BFGS-type method for large-scale unconstrained optimization , 2008, Comput. Math. Appl..

[8]  Ya-Xiang Yuan,et al.  Optimization Theory and Methods: Nonlinear Programming , 2010 .

[9]  Roger Fletcher,et al.  A Rapidly Convergent Descent Method for Minimization , 1963, Comput. J..

[10]  Lenhart K. Schubert.  Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian , 1970 .

[11]  Peter Schlattmann,et al.  Theory and Algorithms , 2009 .

[12]  Andrzej Stachurski.  On the Structure of Variable Metric Updates , 2009 .

[13]  William C. Davidon,et al.  Variable Metric Method for Minimization , 1959, SIAM J. Optim..

[14]  R. Fletcher,et al.  A New Approach to Variable Metric Algorithms , 1970, Comput. J..

[15]  Josef Stoer.  On the convergence rate of imperfect minimization algorithms in Broyden'sβ-class , 1975, Math. Program..

[16]  ON SOLVING A SET OF NONLINEAR EQUATIONS FOR THE DETERMINATION OF STRESSES IN RC RING SECTIONS WITH OPENINGS , 2006 .

[17]  Ya-Xiang Yuan,et al.  Optimization theory and methods , 2006 .

[18]  Mokhtar S. Bazaraa,et al.  Nonlinear Programming: Theory and Algorithms , 1993 .

[19]  R. Kellogg.  A nonlinear alternating direction method , 1969 .

[20]  S. Afriat.  Orthogonal and oblique projectors and the characteristics of pairs of vector spaces , 1957, Mathematical Proceedings of the Cambridge Philosophical Society.

[21]  D. Goldfarb.  A family of variable-metric methods derived by variational means , 1970 .

[22]  Daniel B. Szyld,et al.  The many proofs of an identity on the norm of oblique projections , 2006, Numerical Algorithms.

[23]  M. S. Bazaraa,et al.  Nonlinear Programming , 1979 .

[24]  Identification of an augmented Gurson model parameters for plastic porous media , 2002 .

[25]  Robust identification of an augmented Gurson Model for elasto-plastic porous media , 2006 .

[26]  C. G. Broyden.  The Convergence of a Class of Double-rank Minimization Algorithms 1. General Considerations , 1970 .