# Difference between revisions of "Line search methods"

Line 3: | Line 3: | ||

=Introduction= | =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 <math>f(x)</math>, an initial <math> | + | 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> |

− | = | + | =Step Length= |

− | = | + | |

+ | =Step Direction = | ||

jadklfjlasjfkladsl'''kfdsklf''' | jadklfjlasjfkladsl'''kfdsklf''' | ||

dfadjfkhdakjfhadskj | dfadjfkhdakjfhadskj | ||

Line 13: | Line 14: | ||

[https://www.youtube.com/ Youtube Site] | [https://www.youtube.com/ Youtube Site] | ||

− | = | + | ==Steepest Descent Method== |

+ | ==Newton Method== | ||

+ | ==Quasi-Newton Method== | ||

[[File:Chemicals.jpg]] | [[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]] | ||

− | |||

− | |||

− | |||

=Conclusion= | =Conclusion= | ||

<math>\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}</math> | <math>\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}</math> | ||

− | |||

− | |||

Line 30: | Line 28: | ||

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

## Revision as of 09:31, 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 , an initial

# Step Length

# Step Direction

jadklfjlasjfkladsl**kfdsklf**
dfadjfkhdakjfhadskj
fahdfkjadshf*kahdfjsdk* [1]
Youtube Site

## Steepest Descent Method

## Newton Method

## Quasi-Newton Method

# Conclusion

# 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.