What is the best built-in optimizer in Matlab for this problem?

Question

I need to optimize

$$\sum_{j=1}^p\frac{\sigma^2 \alpha_j^2}{\sigma^2+\delta_j^2\alpha_j^2}-\frac{\left(\displaystyle\sum_{j=1}^p \displaystyle\frac{\sigma^2 ( \delta_j^2 \alpha^2 - \theta\alpha_j)}{\sigma^2+\delta_j^2\alpha_j^2}\right)^2}{(p^2\sigma^2+p \theta^2) - \displaystyle\sum_{j=1}^p \frac{(\sigma^2+\theta \delta_j\alpha_j)^2}{\sigma^2+\delta_j^2\alpha_j^2}}$$

it terms of $\alpha$ where

• $\alpha$ is a vector of four parameters,
• $\sigma$ is a constant,
• $\delta$ is the product of $\alpha$ and another vector which is fixed in value, and
• $\theta$ is the sum of the $\delta$'s.

I think fminunc would be most suitable given the problem in non-linear, and I don't want to use any constraints (unless I have to).

Just wondered if fminunc is indeed most suited to my problem?

In the Matlab documentation, you are invited to use a table to choose the best solver (link); it seems to suggest quadprog but doesn't even mention fminunc?

Thanks

Answer 1

For this problem you are interested in unconstrained optimization and you have a smooth nonlinear problem objective (assuming that the denominator does not blow up). From the table in the Matlab documentation, you have two choices: either fminunc or fminsearch are appropriate. (Also, the table indicates that quadprog is appropriate but only for a quadratic objective.)