Given the following optimization problem:
$\min_{w_t} |w_t - w_{t-1}|^T\gamma$
s.t. $w_t^T\phi \leq 0.15$
where $w_t, w_{t-1}, \phi, \gamma \in \mathbb{R}^{N\times 1}$ and $\phi > 0, \gamma > 0$
I am trying to find closed-form solution for this problem. However, before I attempt to write the Lagrangian and proceed to use the KKT conditions to solve for the solution, I believe I need to reformulate the optimization problem (as we cannot take derivatives of the absolute-value function required for the stationarity condition of KKT).
Any help, guidance or reference on how I can proceed to reformulate the problem and subsequently solve for the analytical solution will be very much appreciated.
Thank you in advance.
Introduce variable $x\in \mathbb{R}^{N}$ to represent the absolute value, and the resulting linear programming problem is to minimize $x^T \gamma$ subject to \begin{align} x &\ge w_t - w_{t-1} \\ x &\ge -w_t + w_{t-1} \\ w_t^T\phi &\le 0.15 \end{align}
In standard form, minimize $\gamma^T x$ subject to \begin{align} x - w_t &\ge - w_{t-1} \\ x + w_t &\ge w_{t-1} \\ -\phi^T w_t &\ge -0.15 \end{align}
The dual problem is to maximize $-w_{t-1}^T\lambda+w_{t-1}^T\theta-0.15\psi$ subject to \begin{align} \lambda + \theta &= \gamma \\ -\lambda + \theta - \phi \psi &= 0 \\ \lambda &\ge 0\\ \theta &\ge 0 \\ \psi &\ge 0 \end{align}
Alternatively, introduce variables $x^+, x^-\in \mathbb{R}^{N}$, and minimize $(x^++x^-)^T \gamma$ subject to \begin{align} x^+ - x^- &= w_t - w_{t-1} \\ x^+ &\ge 0 \\ x^- &\ge 0 \\ w_t^T\phi &\le 0.15 \end{align}