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Regularization Parameter Selection Methods in Parallel MR Image Reconstruction
Published in Joseph Suresh Paul, Raji Susan Mathew, Regularized Image Reconstruction in Parallel MRI with MATLAB®, 2019
Joseph Suresh Paul, Raji Susan Mathew
Following estimation of the trace value, the UPRE functional is computed using Equation (3.16). The two methods used to locate the minimizer of the UPRE functional are the exhaustive search and golden section search [80]. In the exhaustive search method, the parameter values are swept through all λ points to find the optimal parameter value corresponding to the minimum of the cost function. The golden section search method progressively reduces the interval for locating the minimum. This is based on the idea that the minimum lies within the interval defined by the two points adjacent to the point with the least value so far evaluated. Compared to the exhaustive search method, the golden section search method is faster as it computes the UPRE functional using a fewer number of search points.
Search Methods
Published in Theodore Louis, Behan Kelly, Introduction to Optimization for Environmental and Chemical Engineers, 2018
Two of the procedures that are reviewed in this chapter are the interval halving (or bisection) and golden section search methods. These two methods can also be employed to obtain the solution to some equations. For example, the interval halving method depends on finding an approximation to a solution of the form f(x) = 0. Initial guesses x0 and x1 are sequenced provided that f(x) is continuous for x0 ≤ x ≤ x1 and the product [( f(x0))( f(x1))] < 0. This guarantees that the curve y = f(x) crosses the x-axis between x0 and x1. A new approximation of f(x) is then calculated at
Numerical Search Techniques in Single-Variable Optimization
Published in William P. Fox, Nonlinear Optimization, 2020
Golden section search is a search procedure that utilizes the golden ratio. To better understand the golden ratio, consider a line segment over the interval that is divided into two separate regions as shown in Figure 4.2. These segments are divided into the golden ratio if the length of the whole line is to the length of the larger part as the length of the larger part is to the length of the smaller part of the line. Symbolically, this can be written as 1r=r1−r
Optimal strategies for members in a two-echelon supply chain over a safe period under random machine hazards with backlogging
Published in Journal of Industrial and Production Engineering, 2022
Brojeswar Pal, Subhankar Adhikari
Our main problem is the minimization of a function for a single variable. A one-dimensional search method can be applied. The Golden-section searches such a method. But the essential criterion for using this method is that the function has to be unimodal in the specified interval. Here, the criterion of unimodality is verified graphically. For Case 1, unimodality of cost function under two approaches namely centralized and manufacturer dominated are verified through Figure 7 and Figure 8 respectively. For Case 2, by Figure 9 and Figure 10 it is verified for two approaches. The Basic advantage of the Golden-section search method there is no requirement of evaluating derivatives while the limitation of this method is that it can be used only when if the function is unimodal in the specified interval.