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Given $L^1$ functional space $V=L^1([a,b])$ and a positive real number $B$. Let $\varphi$ and $g$ be continuous functions defined on $[a,b]$ such that $\varphi(x)\geq 0$ and $g(x)\geq 0$ for all $x\in [a,b]$. Consider the following optimization problem: $$ \min_{f \in V} \hspace{0.5cm} \left\{ \int_a^b \varphi(x) f(x)dx : \int_a^b f(x)dx=B \text{ and } 0\leq f(x)\leq g(x) \forall x\in [a,b] \right\}. $$ (this problem can be considered as a variant of Knapsack Problem with respect to functional space)

I am looking for solution or name of above problem. Any solution methods for similar problems are also welcome!

Leonard Neon
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  • So this is actually a more direct generalization of the continuous knapsack problem, and an exact generalization if $g$ is piecewise continous. In particular, if $g$ takes on $n$-values, it is an exact case of the continous knapsack problem, which is solvable in $O(n)$ time (disregarding time spent computing partial integrals) – Pax Aug 23 '20 at 18:39
  • Thank you for your information. – Leonard Neon Aug 23 '20 at 18:44

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