Quasi-Newton BFGS algorithm

Both difficulties are eliminated by replacing V L by a positive-definite quasi-Newton (QN) approximation B, which is updated using only values of L and V L (See Section 6.4 for a discussion of QN updates.) Most SQP algorithms use Powell s modification (see Nash and Sofer, 1996) of the BFGS update. Hence the QP subproblem becomes... 

There Eire other Hessian updates but for minimizations the BFGS update is the most successful. Hessism update techniques are usually combined with line search vide infra) and the resulting minimization algorithms are called quasi-Newton methods. In saddle point optimizations we must allow the approximate Hessian to become indefinite and the PSB update is therefore more appropriate. 

In quasi-Newton methods, a parametrized estimate of F or G is used initially, then updated at each iterative step. Saved values of q, p can be used in standard algorithms such as BFGS, described above. 

In order to use MATLAB, it is necessary to write an M-file that defines the sum of squares of errors. Then the command fminu is used to calculate the minimum. The default algorithm in MATLAB is the BFGS quasi-Newton method (Ljung, 2007). 

MATLAB The Math Works, Inc., Natick, MA info mathworks.com http //www.mathworks.com SD, DFR " and BFGS quasi-Newton, simplex algorithm, and others for linear and quadratic programming, least squares, etc. 

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