fa.functional analysis – Are there references to functional variants of the Unbounded Knapsack Problem?

Looking for a version of the following problem, extended to solutions in $ell^{infty}(mathbb{N})$


Unbounded Knapsack Problem

$ max_{x_1,…,x_n} sum_{i=1}^n v_ix_i$

$text{ subject to }$

$sum_{i=1}^n w_ix_i leq W$

$w_i,v_i in mathbb{R}^+ ;;forall i$

$x_i in mathbb{N}$


What I’d like to examine is the extended version

“Functional” Unbounded Knapsack Problem

$ max_{x_1,…,x_n} langle v,x rangle$

$text{ subject to }$

$langle w,x rangle leq W$

$w,v in ell^{infty}:w_i,v_i > 0;forall i in mathbb{N}$

$x in mathbb{N}^{infty}$


I’m having a hard time finding any existing literature on this problem (or its continuous cousin defined on $L^{infty}$).

More generally, is there something akin to Calculus of Variations (except we’re optimizing functionals defined on $ell^{infty}$). Is there a variant of the Euler-Lagrange equation for functionals in sequence spaces?