Problem
Is it possible to find a closed-form solution based on the system parameters ($a,b,c,d$ and $h$) for the following convex optimization problem: \begin{align*} \min_{\substack{t \geq 0}} \quad & b \big(\frac{a}{b} \big)^t + a \big(\frac{a}{b} \big)^{-t} + d \big(\frac{c}{d} \big)^t + c \big(\frac{c}{d} \big)^{-t} -ht \\ \text{s.t.} \quad & a > b > 0, \\ & c > d > 0, \\ & h \geq0 \\ \end{align*} I cannot solve it when $h > 0$.
My attempt:
Since the function is convex with respect to $t$, it has a unique global minimum (each local minimum is the global minimum), and its minimum happens at ${t^{\ast}}$ where the derivative function is zero, $f^\prime(t^\ast)=0$. \begin{align} f^\prime(t) =\ln\big(\frac{a}{b} \big) \Big[ b \big(\frac{a}{b} \big)^t - a \big(\frac{a}{b} \big)^{-t} \Big] + \ln\big(\frac{c}{d} \big) \Big[ d \big(\frac{c}{d} \big)^t - c \big(\frac{c}{d} \big)^{-t} \Big] - h \end{align} If $h=0$, then $t^{\ast} = \frac{1}{2}$, and $f(t^{\ast}) = 2 \sqrt{ab} +2 \sqrt{cd}$. I could not proceed further due to high nonlinearity in $f^\prime(t)$ when $h > 0$.
Thanks for your help in advance.