Numerical Optimization
Numerical Optimization
Convergence Rates
Wikipedia
The rate of convergence is \(\displaystyle \lim_{k \rightarrow \infty} \frac{|x_{k+1}-x^*|}{|x_{k}-x^*|^q}=L\)
Iterative methods have the following convergence rates:
- If Q=1 and L=1 we have sublinear convergence.
- If Q=1 and \(\displaystyle L\in(0,1)\) we have linear convergence.
- If Q=1 and \(\displaystyle L=0\) we have superlinear convergence.
- If Q=2 and \(\displaystyle L \leq \infty\) we have quadratic convergence.
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 \(\displaystyle \Vert p \Vert \lt \Delta_k \).
Cauchy Point Algorithms
The Cauchy point \(\displaystyle p_k^c = \tau_k p_k^s\)
where \(\displaystyle p_k^s\) minimizes the linear model in the trust region
\(\displaystyle p_k^s = \operatorname{argmin}_{p \in \mathbb{R}^n} f_k + g_k^Tp \) s.t. \(\displaystyle \Vert p \Vert \leq \Delta_k \)
and \(\displaystyle \tau_k\) minimizes our quadratic model along the line \(\displaystyle p_k^s\):
\(\displaystyle \tau_k = \operatorname{argmin}_{\tau \geq 0} m_k(\tau p_k^s)\) s.t. \(\displaystyle \Vert \tau p_k^s \Vert \leq \Delta_k \)
This can be written explicitly as \(\displaystyle p_k^c = - \tau_k \frac{\Delta_k}{\Vert g_K \Vert} g_k\) where \(\displaystyle \tau_k =
\begin{cases}
1 & \text{if }g_k^T B_k g_k \leq 0;\\
\min(\Vert g_k \Vert ^3/(\Delta_k g_k^T B_k g_k), 1) & \text{otherwise}
\end{cases}
\)
Conjugate Gradient Methods
The goal is to solve \(\displaystyle Ax=b\) or equivalently \(\displaystyle \min \phi(x)\) where \(\displaystyle \phi(x)=(1/2)x^T A x - b^Tx\)
.
The practical CG algorithm will converge in at most \(\displaystyle n\) steps.
Definitions
Vectors \(\displaystyle \{p_i\}\) are conjugate w.r.t SPD matrix \(\displaystyle A\) if \(\displaystyle p_i^T A p_j = 0\) for \(\displaystyle i \neq j\)
- Notes
- \(\displaystyle \{p_i\}\) are linearly independent
Algorithms
Basic idea:
- Find a conjugate direction \(\displaystyle p_k\)
- Take the step size \(\displaystyle \alpha_k\) which minimizes along \(\displaystyle \phi(x)\)