Let $\Phi$ denote the standard Gaussian CDF. Given $a \in \mathbb R^n$, $\alpha > 0$, and $0 < \sigma_1 \le \ldots \le \sigma_n$, find $w \in \mathbb R^n$ which minimizes $\gamma:=\Phi\left(\frac{w^Ta+\alpha\|w\|_1}{\sqrt{\sum{\sigma_i^2 w_i^2}}}\right)-\Phi\left(\frac{w^Ta}{\sqrt{\sum_{i=1}^n{\sigma_i^2 w_i^2}}}\right)$.
Any help welcome!
Observation
Noting that $\alpha\|w\|_1 :=\max_{\|z\|_\infty \le \alpha} w^Tz$ and $\Phi$ is an increasing function, one has $$ \begin{split} \gamma &= \min_{w \in \mathbb R^n}\Phi\left(\frac{w^Ta+\max_{\|z\|_\infty \le \alpha}w^Tz}{\sqrt{\sum{\sigma_i^2 w_i^2}}}\right)-\Phi\left(\frac{w^Ta}{\sqrt{\sum_{i=1}^n{\sigma_i^2 w_i^2}}}\right)\\ &=\min_{w \in \mathbb R^n}\max_{\|z\|_\infty \le \alpha}\left(\Phi\left(\frac{w^T(a+z)}{\sqrt{\sum{\sigma_i^2 w_i^2}}}\right)-\Phi\left(\frac{w^Ta}{\sqrt{\sum_{i=1}^n{\sigma_i^2 w_i^2}}}\right)\right)\\ &=\max_{\|z\|_\infty \le \alpha}\min_{w \in \mathbb R^n}\left(\Phi\left(\frac{w^T(a+z)}{\sqrt{\sum{\sigma_i^2 w_i}}}\right)-\Phi\left(\frac{w^Ta}{\sqrt{\sum_{i=1}^n{\sigma_i^2 w_i^2}}}\right)\right)\\ &= \max_{\|z\|_\infty \le \alpha}\;\min_{\sum_{i=1}^n \sigma_i^2w_i^2 \le 1}\left(\Phi\left(w^T(a+z)\right)-\Phi\left(w^Ta\right)\right) \end{split} $$ where the last but one equality is Sion's minimax inequality.