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Unformatted text preview: ined tolerance (a small number!) [For you: How do you know how many correct digits you have in your solution?] RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 31 Overview of Line-Search and Newton Type Methods of Unconstrained Optimization We first describe the Newton method for the one-dimensional unconstrained optimization problem min f (x ) x where we assume that the first and second order derivatives of f exist Assume we have an approximation x k of the optimal solution x * and we want to compute a “better” approximation x k +1 . The Taylor series expansion around x k is given by 1 f (x k + h ) = f (x k ) + f ¢(x k )h + f ¢¢(x k )h 2 + O(h 3 ) 2 where h is some small number. For h is small enough, we solve the optimization problem 1 min f (x k ) + f ¢(x k )h + f ¢¢(x k )h 2 h 2 RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 32 1 min f (x k ) + f ¢(x k )h + f ¢¢(x k )h 2 h 2 This is a simple quadratic optimization problem in h. By taking derivatives with respect to h we have f ¢(x k ) + f ¢¢(x k )h = 0 Solving for h we obtain h =- f ¢(x k ) f ¢¢(x k ) Therefore, the new approximation x k +1 becomes x k +1 = x k + h = x k - f ¢(x k ) f ¢¢(x k ) This is the Newton method for the one-dimensional unconstrained optimization problem above RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 33 N-Dimensional Unconstrained Optimization The Newton method is easily extended to N-dimensional problems and then takes the form8 x k +1 = x k + h = x k - [2 f (x k )]-1 f (x k ) where x k +1 , x k are N dimensional vectors, and f (x k ) and 2 f (x k ) are the gradient and the Hessian of f at x k , respectively Note: You should not calculate [2 f (x k )]-1 explicitly and then multiply it with f (x k ). [For you: Why?] Rather, you should solve the linear system for h [2 f (x k )]-1 h = -f (x k ) RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 34 Line Search Strategies The Newton method is a so-called line search strategy: After the k-th step, x k is given and the (k+1)-th approximation is calculated according to the iterative scheme x k +1 = x k + g pk where pk Î R N is the search direction chosen by the algorithm [Of course, in the case of the Newton method the search direction is chosen to be pk = -[2 f (x k )]-1 f (x k ) and g = 1 .] Other search directions lead to algorithms with different properties RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 35 Example: Method of steepest descent the search direction is chosen as9 pk = -f (x k ) Steepest descent requires the first-order derivatives of the function f, and not second-order derivatives as the Newton method Therefore, a steepest descent iteration is computationally cheaper to perform than a Newton iteration. RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 36 Convergence Steepest descent and the Newton method have different convergence properties. The rate of convergence to...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.

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