Difference between revisions of "Line search methods"

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An algorithm is a line search method if it seeks the minimum of a defined nonlinear function by selecting a reasonable direction vector that, when computed iteratively with a reasonable step size, will provide a function value closer to the absolute minimum of the function. Varying these will change the "tightness" of the optimization. For example, given the function <math>f(x)</math>, an initial <math>x_k</math> is chosen. To find a lower value of <math>f(x)</math>, the value of <math>x_{k+1}</math> is increased by the following iteration scheme
 
An algorithm is a line search method if it seeks the minimum of a defined nonlinear function by selecting a reasonable direction vector that, when computed iteratively with a reasonable step size, will provide a function value closer to the absolute minimum of the function. Varying these will change the "tightness" of the optimization. For example, given the function <math>f(x)</math>, an initial <math>x_k</math> is chosen. To find a lower value of <math>f(x)</math>, the value of <math>x_{k+1}</math> is increased by the following iteration scheme
  
[[File:CodeCogsEqn.gif]]
+
[[File:CodeCogsEqn.gif]],
 +
 
 +
in which <math>\alpha_k</math> is a positive scalar known as the step length and <math>p_k</math> defines the step direction.
  
 
=Step Length=
 
=Step Length=
  
 
=Step Direction =
 
=Step Direction =
jadklfjlasjfkladsl'''kfdsklf'''
+
 
dfadjfkhdakjfhadskj
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fahdfkjadshf''kahdfjsdk'' [1]
+
[https://www.youtube.com/ Youtube Site]
+
  
 
==Steepest Descent Method==
 
==Steepest Descent Method==
 
==Newton Method==
 
==Newton Method==
 
==Quasi-Newton Method==
 
==Quasi-Newton Method==
[[File:Chemicals.jpg]]
+
 
 
[[File:48StatesTSP.png|frame|Solution to 48 States Traveling Salesman Problem]]
 
[[File:48StatesTSP.png|frame|Solution to 48 States Traveling Salesman Problem]]
  

Revision as of 10:54, 24 May 2015

Author names: Elizabeth Conger
Steward: Dajun Yue and Fengqi You

Contents

Introduction

An algorithm is a line search method if it seeks the minimum of a defined nonlinear function by selecting a reasonable direction vector that, when computed iteratively with a reasonable step size, will provide a function value closer to the absolute minimum of the function. Varying these will change the "tightness" of the optimization. For example, given the function f(x), an initial x_k is chosen. To find a lower value of f(x), the value of x_{k+1} is increased by the following iteration scheme

CodeCogsEqn.gif,

in which \alpha_k is a positive scalar known as the step length and p_k defines the step direction.

Step Length

Step Direction

Steepest Descent Method

Newton Method

Quasi-Newton Method

Solution to 48 States Traveling Salesman Problem

Conclusion

\begin{bmatrix} G(x,y) & 0 & -A(x)^T \\ 0 & Y & W \\ A(x) & -I & 0 \end{bmatrix} \begin{bmatrix} \Delta x \\ \Delta s \\ \Delta y \end{bmatrix} = \begin{bmatrix} -\nabla f(x) + A(x)^T y \\ \mu e - W Y e \\ -g(x) + s \end{bmatrix}


References

1. Sun, W. & Yuan, Y-X. (2006) Optimization Theory and Methods: Nonlinear Programming (Springer US) p 688.

2. Anonymous (2014) Line Search. (Wikipedia). http://en.wikipedia.org/wiki/Line_search.

3. Nocedal, J. & Wright, S. (2006) Numerical Optimization (Springer-Verlag New York, New York) 2 Ed p 664.