I want to find the analytical minimum 'x_opt=argmin(x)' of the following function:
f(x) = alpha * |c + x| + beta * x ^ 2
where x is a real number (x is_element_of R), c is a real constant (c is_element_of R), alpha and beta are positive real constants (alpha is_element_of R+), beta is_element_of R+), |?| is the absolute value function and ^ is the power function.
Looks simple, but the absolute value function makes it somewhat tricky. As already mentioned, i want to find the solution to this minimization problem analytically, not numerically.
I managed to split the optimization up according to the three cases x < -c, x = -c, x > -c, and solve each case separately analytically (by setting the first derivative to zero).
For the three cases I have now the solutions x_opt = alpha/(2*beta) [for x < -c], x_opt = -c [for x = -c], and x_opt = -alpha/(2*beta) [for x > -c]. But how to 'combine' these solutions now to get the solution 'x_opt' (as a function of 'x') ?
So i would need a function 'phi(x)' which delivers me 'x_opt' for a given x, 'phi(x) = argmin f(x)'. How does phi(x) look like ?