Does this optimization problem have a unique solution?

Question

I have come across a seemingly simple optimization problem. Let $h\in L^1(0,1)$; consider the optimization problem \begin{equation} \begin{cases} \min \int_0^1\int_0^t h(s)dsdt\\ \text{s.t. } \int_0^1h(t)(1-t)>1/2, \quad 0\le h(t)\le 2. \end{cases} \end{equation} There seems to be infinitely many local minimizers to this problem. The first-order optimality condition is derived as \begin{equation} \int_t^11\,ds-\mu_1(1-t)+\mu_2(t)-\mu_3(t)=0, \end{equation} where $\mu_1$, $\mu_2(t)$, and $\mu_3(t)$ are non-negative Lagrange multipliers. The only legitimate value for these multipliers will be $\mu_1=1$, $\mu_2(t)=\mu_3(1)=0$; but, this basically means no information except that $\int_0^1h_{opt}(t)(1-t)=1/2$.

Answer 1

Your objective coincides with the left-hand side of your integral constraint. Hence, your problem does not possess any minimizer, since the infimal value is $1/2$ but it cannot be achieved.