I have an optimization problem of the form
$\max_\gamma \langle f\circ\gamma,w_f\rangle\langle g\circ\gamma,w_g\rangle$
where $f\circ\gamma$ is the composition of $f$ and $\gamma$ and the inner product is defined as
$\langle f\circ\gamma,w_f\rangle = \int (f\circ\gamma)(t) w_f(t)\,dt$
In other words I am trying to find a $\gamma$ that maximizes the product of the two inner products above. I know one could pursue a gradient approach, which would be slow. Is there a more direct optimization method?
If $f$ and $g$ are differentiable, you could do something like the following.
Define $J(\gamma)$ as your objective function. Then, the variation of $J$ in direction $h$ is given by \begin{align*}J'(\gamma; h) &= \int f'(\gamma(t)) \, h(t) \, w_f(t) \, \mathrm d t \, \int g(\gamma(t)) \, w_g(t) \, \mathrm d t + \int f(\gamma(t)) \, w_f(t) \, \mathrm d t \, \int g'(\gamma(t)) \, h(t) \, w_g(t) \, \mathrm d t\\ &= \int \Bigg\{\Big( \int g(\gamma(s)) \, w_g(s) \, \mathrm d s \Big) f'(\gamma(t)) \, w_f(t) + \Big(\int f(\gamma(s)) \, w_f(s) \, \mathrm d s\Big) \, g'(\gamma(t)) \, w_g(t) \, \Bigg\} \, h(t) \mathrm d t\\ \end{align*} Now, this has to be zero for all $h$. Hence, for all $t$, you get the equation $$\Big( \int g(\gamma(s)) \, w_g(s) \, \mathrm d s \Big) f'(\gamma(t)) \, w_f(t) + \Big(\int f(\gamma(s)) \, w_f(s) \, \mathrm d s\Big) \, g'(\gamma(t)) \, w_g(t) = 0$$ Now, it depends on the functions how this equation could be solved.
Maybe, this is of some use.