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Newton’s Method

Newton’s Method. Given an objective function f(x), Min. S.t. where Q is the Hessian matrix of  . Solution: The Newton’s direction is given by Iteration formula:

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Newton’s Method

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  1. Newton’s Method Given an objective function f(x), Min. S.t. where Q is the Hessian matrix of . Solution: The Newton’s direction is given by Iteration formula: Where k+1 is determined by minimizing  along the Newton’s direction. Note that if Q is replaced by identity matrix, then the above formulae are reduced to the steepest descent method.

  2. Quasi-Newton or Variable Metric Algorithms: • By Taylor series expansion: • or in a more compact form, , where , and • Solution of the above equation is: • A typical variable-metric algorithm with an inverse Hessian update may be stated as: where

  3. Rank-One Updates: • Rank-Two Updates: where

  4. Davidon-Fletcher-Powell’s (DFP) update formula: • Setting k =0, k=0 for all k, • BFGS formula: • Setting k =1, k=1 for all k, • We can also get the update for the Hessian matrix approximations:

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