I'm trying to solve the following problem: Suppose $F:\mathbb{R_+}\to\mathbb{R_+}$ is a given decreasing function with $F(0)=1$ and $F(\infty)=0$. \begin{align*} &\min{\int_0^{\infty} F(t) e^{-M(t)}}\textrm{d}t\\ {\textrm{st.}} & \int_0^{\infty} F(t) e^{-M(t)} M'(t) \textrm{d}t \leq \alpha, \end{align*} with $\alpha\in(0,1)$. The problem is finding $M:\mathbb{R_+}\to\mathbb{R_+}$ with the boundary condition $M(0)=0$ such that the objective is minimized and the constraint is satisfied. Any input would be appreciated!
Update: When $F(t)=e^{-t}$, I conjecture that the solution is the linear function $M(t)=\frac{\alpha}{1-\alpha}t$, but cannot prove this.
Solve the problem without the inequality constraint. If you find a minimal solution that satisfies the constraint, you’re done. If not, the minimum must lie on the boundary, so you can replace the inequality by an equality and solve the constrained problem using a Lagrange multiplier.