Practical methods of optimization; (2nd ed.)
Practical methods of optimization; (2nd ed.)
Optimal conditioning and convergence in rank one quasi-Newton updates
SIAM Journal on Numerical Analysis
Convergence of quasi-Newton matrices generated by the symmetric rank one update
Mathematical Programming: Series A and B
Sizing and least-change secant methods
SIAM Journal on Numerical Analysis
Measures for symmetric rank-one updates
Mathematics of Operations Research
Remark on “Algorithm 500: Minimization of Unconstrained Multivariate Functions [E4]”
ACM Transactions on Mathematical Software (TOMS)
Testing Unconstrained Optimization Software
ACM Transactions on Mathematical Software (TOMS)
A Modified Rank One Update Which Converges Q-Superlinearly
Computational Optimization and Applications
Improved Hessian approximation with modified secant equations for symmetric rank-one method
Journal of Computational and Applied Mathematics
Computers & Mathematics with Applications
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Two basic disadvantages of the symmetric rank one (SR1) update are that the SR1 update may not preserve positive definiteness when starting with a positive definite approximation and the SR1 update can be undefined. A simple remedy to these problems is to restart the update with the initial approximation, mostly the identity matrix, whenever these difficulties arise. However, numerical experience shows that restart with the identity matrix is not a good choice. Instead of using the identity matrix we used a positive multiple of the identity matrix. The used positive scaling factor is the optimal solution of the measure defined by the problem--maximize the determinant of the update subject to a bound of one on the largest eigenvalue. This measure is motivated by considering the volume of the symmetric difference of the two ellipsoids, which arise from the current and updated quadratic models in quasi-Newton methods. A replacement in the form of a positive multiple of the identity matrix is provided for the SR1 update when it is not positive definite or undefined. Our experiments indicate that with such simple initial scaling the possibility of an undefined update or the loss of positive definiteness for the SR1 method is avoided on all iterations.