Preliminaries ‘steepest descent method’ “Newton’s method” Conjugate Gradient1 We have learned ‘steepest descent method’ and “Newton’s method”. The main advantage of Newton’s method is the speed, it converges quickly. And the main advantage of the steepest descent method guarantees to converge to a local minimum. But the limit of Newton’s method is that it needs too many resources for both computation and storage when the number of parameters is large....
Preliminaries ‘steepest descent algorithm’ Linear Algebra Calculus 1,2 Newton’s Method1 Taylor series gives us the conditions for minimum points based on both first-order items and the second-order item. And first-order item approximation of a performance index function produced a powerful algorithm for locating the minimum points which we call ‘steepest descent algorithm’. Now we want to have an insight into the second-order approximation of a function to find out whether there is an algorithm that can also work as a guide to the minimum points....
Preliminaries ‘An Introduction to Performance Optimization’ Linear algebra Calculus 1,2 Direction Based Algorithm and a Variation1 This post describes a direction searching algorithm(\(\mathbf{x}_{k}\)). And its variation gives a way to estimate step length (\(\alpha_k\)). Steepest Descent To find the minimum points of a performance index by an iterative algorithm, we want to decrease the value of the performance index step by step which looks like going down from the top of the hill....
Preliminaries Nothing Performance Optimization1 Taylor series had been used for analyzing the performance surface and locating the optimum points of a certain performance index. This short post is a brief introduction to performance optimization and the following posts are the samples of three optimization algorithms categories: ‘Steepest Descent’ “Newton’s Method” ‘Conjugate Gradient’ Recall the analysis of the performance index, which is a function of the parameters of the model....
Preliminaries Linear algebra Calculus 1,2 Taylor series Quadratic Functions1 Quadratic function, a type of performance index, is universal. One of its key properties is that it can be represented in a second-order Taylor series precisely. \[ F(\mathbf{x})=\frac{1}{2}\mathbf{x}^TA\mathbf{x}+\mathbf{d}\mathbf{x}+c\tag{1} \] where \(A\) is a symmetric matrix(if it is not symmetric, it can be easily converted into symmetric). And recall the property of gradient: \[ \nabla (\mathbf{h}^T\mathbf{x})=\nabla (\mathbf{x}^T\mathbf{h})=\mathbf{h}\tag{2} \] and \[ \nabla (\mathbf{x}^TQ\mathbf{x})=Q\mathbf{x}+Q^T\mathbf{x}=2Q\mathbf{x}\tag{3} \]...