I would like to find the analytical form of the point which minimizes the following function:$$f(x_T) = \frac{1}{T} a_1 (x_T-x_0)^2 + a_2 |x_T-x_0| + T a_3 + \sum_{i=1}^M p_i \left[b_{1i} (x_T - l_i)^2 + b_{2i} |x_t - l_i| + b_{3i} \right]$$where $T,a_1,x_0,a_2,a_3,p_i,b_{1i},l_i,b_{2i},b_{3i}$ are constants, $T,a_1,x_0,a_2,p_i,b_{1i},l_i,b_{2i}$ are greater than zero, and M is a positive index. The function is continuous, convex, and piecewise differentiable. Any ideas on how to find the analytical form of the point, $x^*_T$, which minimizes $f(x_T)$?
Analytical form of the minimum of a function with absolute values
optimization;convex optimization