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

Painless Conjugate Gradient
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 \(\displaystyle \phi(x)\)

Practical CG method:

Code

Below is code for the practical CG method.

x = x0;
r = A*x - b;
p = -r;
k = 1;
while(norm(r) > tole && k < size(x,2))
   Ap = A*p;
   alpha = (r'*r)/(p'*Ap);
   x = x + alpha * p;
   rn = r + alpha * Ap;
   beta = (rn'*rn)/(r'*r);
   p = -rn + beta * p;
   r = rn;
   k = k+1;
end

Theorems

Given a set of conjugate directions \(\displaystyle \{p_0,...,p_{n-1}\}\) we can generate a sequence of \(\displaystyle x_k\) with
\(\displaystyle x_{k+1}=x_k + \alpha_k p_k\) where \(\displaystyle \alpha_k = -\frac{r_k^T p_k}{p_k^T A p_k}\) minimizes \(\displaystyle \phi(x)\)

Theorem: For an \(\displaystyle x_0\), the sequence \(\displaystyle x_k\) converges to the solution \(\displaystyle x^*\) in at most n steps.
Theorem: At step k, the residual \(\displaystyle r_k\) is orthogonal to \(\displaystyle \{p_0,...p_{k-1}\}\) and the current iteration \(\displaystyle x_{k}\); minimizes \(\displaystyle \phi(x)\) over the set \(\displaystyle \{x_0 + span(p_0,...,p_{k-1})\}\)

Convergence Rate

The convergence rate can be estimated by:
\(\displaystyle \min_{P_k} \max_{1 \leq i \leq k} [1 + \lambda_i P_k(\lambda_i)]^2\)

Resources

Numerical Optimization by Nocedal and Wright (2006)