OPTIMIZATION.pptx

Approaches in the sense that the optimal solution is

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approaches in the sense that the optimal solution is bracketed by these boundaries. Newton’s method is an open (instead of bracketing) approach, where the optimum of the one-dimensional function f(x) is found using an initial guess of the optimal value without the need for specifying lower and upper boundary values for the search region.

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NEWTON’S METHOD Newton’s method is an open approach to find the minimum or the maximum of a function f(x). It is very similar to the Newton-Raphson method to find the roots of a function such that f(x) = 0. Since the derivative of the function f(x), f(x)=0 at the functions maximum and minimum, the minima and the maxima can be found by applying the Newton-Raphson method to the derivative, essentially obtaining
NEWTON’S METHOD We caution that before using Newton’s method to determine the minimum or the maximum of a function, one should have a reasonably good estimate of the solution to ensure convergence, and that the function should be easily twice differentiable.

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GOLDEN SECTION SEARCH METHOD The golden-section search is a technique for finding the extremum (minimum or maximum) of a strictly unimodal functionby successively narrowing the range of values inside which the extremum is known to exist. The technique derives its name from the fact that the algorithm maintains the function values for triples of points whose distances form a golden ratio. The algorithm is the limit of Fibonacci search (also described below) for a large number of function evaluations. Fibonacci search and golden-section search were discovered by Kiefer (1953) (see also Avriel and Wilde (1966)).
MULTIDIMENSIONAL DIRECT SEARCH METHOD Pattern search (also known as direct search, derivative-free search, or black-box search) is a family of numerical optimization methods that does not require a gradient. As a result, it can be used on functions that are not continuous or differentiable. One such pattern search method is "convergence" (see below), which is based on the theory of positive bases. Optimization attempts to find

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