Optimization of convex integral function

Question

I have the following constrained optimization with the integral objective function $$ \min_{x_i\in D} \int_{t_1}^{t_2} \frac 1 {t - \sum\limits_{i = 1}^N x_i f_i (t)} \, dt $$ where $t - \sum\limits_{i = 1}^N x_i f_i(t) > 0 $ and $f_i(t)$ are polynomial functions. $D$ is a convex set. How can I solve the optimization problem?

Answer 1

Do not know whether it will work or not, but you can try this way-

Step-$1$: $$\frac{d}{dx_i}\int_{t_1}^{t_2}g(x_i,t)dt=\int_{t_1}^{t_2}\frac{\partial}{\partial x_i}g(x_i,t)dt+\frac{dt_2}{dx_i}\times g(x_i,t_2)-\frac{dt_1}{dx_i} \times g(x_i,t_1).$$

Step-$2$: At $x_i =x_i^*$, the derivative will be zero.

Step-$3.a$: Project $x_i^*$ on $D$ if $x_i^* \notin D$. You will find a unique point because $D$ is a convex set (but also the closeness of the set is required).

Step-$3.b$: No need to do anything further, if $x_i^* \in D$. $x_i^*$ is the answer.