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[64] Luo YZ, Tang GJ, Zhou LN, Hybrid approach for solving systems of nonlinear equations using chaos optimization and quasi-Newton method. Applied Soft Computing 8, 1068-1073, Elsevier 2008
Second, Figure 4.10 suggests that AD exhibits local quadratic convergence behavior, while BFGS yields, as expected, only super-linear convergence which reflects the different natures of the Newton-type and the quasi-Newton methods. Besides, the importance of the merit function and backtracking process is evident.
Newton Raphson's method is less efficient than Quasi-Newton method. In Newton Raphson's method, the Jacobian matrix has to be evaluated for every iteration but with the Quasi-Newton method, a single Jacobian matrix is determined and thus used in the iteration.
In our algorithm, we introduced an efficient limited-memory quasi-Newton method, L-BFGS,to promote efficiency and reduce computational complexity for unconstrained optimization problemswith many variables, which updates with second-order derivativeinformation inferred from observed changes in the gradient.
Ma, "A new smoothing quasi-newton method for nonlinear complementarity problems," Applied Mathematics and Computation, vol.
We obtain a local solution iteratively by using the combination of penalty function method and quasi-Newton method, which is effective to solve the equality constrained optimization problem [26].
So far, as that done in the quasi-Newton method, we can employ the following updating technique due to BFGS:
The ordinary quasi-Newton method for solving (1) generates a sequence {[x.sub.k]} by the following iterative scheme:
The validity of this approach was confirmed by comparing modified and unmodified versions of the quasi-Newton method within the ASM framework, with the model-trust region method clearly outperforming both versions of the quasi-Newton approaches.
However, the most critical part of such quasi-Newton method is storing the full matrix of the approximate Jacobian, which can be a very expensive tas k as the dimension of systems increases [6].
and Barker, R.D., 1996, Rapid least squares inversions of apparent resistivity pseudosections by a quasi-newton method, Geophys.
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