Numerical Optimization: Difference between revisions
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** Find a region where you trust your model accurately represents your objective function. | ** Find a region where you trust your model accurately represents your objective function. | ||
** Take a step. | ** Take a step. | ||
<br> | |||
Variables: | |||
* <math>f</math> is your objective function. | |||
* <math>m_k</math> is your quadratic model at iteration k. | |||
* <math>x_k</math> is your point at iteration k. | |||
Your model is <math>m_k(p) = f_k + g_k^T p + \frac{1}{2}p^T B_k p</math> | |||
where <math>g_k = \nabla f(x_k)</math> and <math>B_k</math> is a symmetric matrix.<br> | |||
At each iteration, you solve a constrained optimization subproblem to find the best step <math>p</math>.<br> | |||
<math>\min_{p \in \mathbb{R}^n} m_k(p)</math> such that <math>\Vert p \Vert < \Delta_k </math>. | |||
==Resources== | ==Resources== | ||
* [https://link.springer.com/book/10.1007%2F978-0-387-40065-5 Numerical Optimization by Nocedal and Wright (2006)<br> | * [https://link.springer.com/book/10.1007%2F978-0-387-40065-5 Numerical Optimization by Nocedal and Wright (2006)<br> | ||
Revision as of 19:50, 31 October 2019
Numerical Optimization
Line Search Methods
Basic idea:
- For each iteration
- Find a direction \(\displaystyle p\).
- Then find a step length \(\displaystyle \alpha\) which decreases \(\displaystyle f\).
- Take a step \(\displaystyle \alpha p\).
Trust Region Methods
Basic idea:
- For each iteration
- Assume a quadratic model of your objective function near a point.
- Find a region where you trust your model accurately represents your objective function.
- Take a step.
Variables:
- \(\displaystyle f\) is your objective function.
- \(\displaystyle m_k\) is your quadratic model at iteration k.
- \(\displaystyle x_k\) is your point at iteration k.
Your model is \(\displaystyle m_k(p) = f_k + g_k^T p + \frac{1}{2}p^T B_k p\)
where \(\displaystyle g_k = \nabla f(x_k)\) and \(\displaystyle B_k\) is a symmetric matrix.
At each iteration, you solve a constrained optimization subproblem to find the best step \(\displaystyle p\).
\(\displaystyle \min_{p \in \mathbb{R}^n} m_k(p)\) such that \(\displaystyle \Vert p \Vert \lt \Delta_k \).
Resources
- [https://link.springer.com/book/10.1007%2F978-0-387-40065-5 Numerical Optimization by Nocedal and Wright (2006)