I am dealing with an optimization problem of the form: $$x^* = \mathop{\text{argmin}}_{x\in \mathbb{R}^+}\left( x^2 - c_1\log x^2 + r_1(x - c_2)^2\right), $$ where $r_1, c_1, c_2$ are constants.
So could anyone gives some advice of this form? No need to analytic form, numeric solution is also appreciated. Thanks in advance!
If $c_1,r_1 \ge 0$ then $f''(x) >0$ and the (unique) root of $f'(x)$ will be a global minimizer. Also, you can compute the roots of $f'(x)$ and pick the one in $\mathbb{R}_+$. So, if $c_1>0$ and $r_1\ge 0$, the answer is $$ x^* = \dfrac{c_2 r_1 + \sqrt{c_2^2 r_1^2+4 c_1 r_1 + 4 c_1}}{2(1+r_1)}. $$