How to reformulate an objective function with absolute

Question

I have the following optimization problem:

$min_{w} \sum^{N}_{i=1} |\mu_i| \cdot log(|w_i|) $ s.t.

$\sqrt{w' \cdot \Sigma \cdot w} \leq 0.05$

$w_i > 0 $ if $\mu_i > 0$

$w_i < 0 $ if $\mu_i < 0$

$\sum^{N}_{i=1} |w_i| = 1$

The optimization algorithm that I have chosen is SQP (for reasons not discussed here). However, this algorithm requires smooth objective functions that are continous and differentiable. As I have the absolute and log in my function, this requirement seems to be violated.

I understand that then the general practice is to reformulate this problem. I am however puzzled how to do this. How could this problem be reformulated? Where do you start?

Answer 1

I am assuming here that $\mu_i$ is constant. Here is a possible formulation:

Let $s_i = \text{sign}(\mu_i)$ and $\varepsilon=10^{-6}$

NLP: $$\begin{align} \min & \sum_i |\mu_i| \log(\varepsilon+s_i w_i)\\ & w'Qw \le 0.05^2\\ & \sum_i s_i w_i = 1 \\ & w_i \le 0 && \forall i| s_i=-1\\ & w_i \ge 0 && \forall i|s_i=+1 \end{align}$$

Notes:

• $Q$ is hopefully positive definite.
• An initial point could be $w^0_i = s_i/n$.
• $w_i \le 0$ and $w_i \ge 0$ are bounds (opposed to general constraints).
• This is non-convex so you may get local optimal solutions when using a local solver.
• With some random data I see variables $w_i$ assuming an optimal value of zero. I.e. the problem can be interpreted to be unbounded. This may be an ill-posed problem.
• Maximization of this objective may make more sense. (No longer unbounded and now also convex).