Unconstrained Minimization: Backtracking Line Search & Gradient Descent

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<a href="http://www.searching-eye.com/sanjeevs.html" target="_blank">SANJEEV SHARMA</a>: 3rd Dec 2010. <a href="http://www.searching-eye.com/sanjeevsharma/co/cco/" target="_blank">CCO-10/11</a>: P-002, Section-2: Unconstrained Minimization: Backtracking Line Search & Gradient Descent. <br> <br> Contents: Exact Line Search; Inexact Line Search; Backtracking Line Search; Gradient Descent. <br><br> Methods for solving the unconstrained minimization problems include Descent Algorithms. These are iterative solvers which iterates between the two steps to find the solution. Step-1 invloves finding a search direction (&delta;x) and step-2 involves finding a step-length <i>t</i> to move in the direction &delta;x. The step-length if found using the line search methods. There are two kinds of line search algorithms, the exact & inexact line search. Exact line search finds the step lenght for which the function f(x+<i>t</i>&delta;x) is minimized i.e. <i>t</i>=argmin_{(s>0)}f(x+s&delta;x). The inexact line search just finds the step length such that the function f(x+<i>t</i>&delta;x) is approximately minimized. The most popular one is the backtracking line search algorithm which depends on two constants, &alpha; & &beta; . Gradient descent is an algorithm which uses the &delta;x=-&nabla;f(x), i.e. the direction is the direction of maximum decrease of f(x). It uses the line search (exact or backtracking) to find the step-length to move in this direction. The performance of the algorithm is dependent on the sublevel sets of f(x) near optimum. This presentation shows how to practically apply and solve the Backtracking Line Search Algorithm.
Channels: CCO-10/11
Tags: Backtracking  Line  Search,  Gradient  Descent,  Exact  Line  Search

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