I am reading a paper where the author derives the following Lagrangian dual problem :

$min_v \int_R \frac{1}{4} \frac{\beta²}{v-2||x||}dx+v\;\;\;\text{s.t.}\;\;\;v\geq 2||x||\;\;\;\forall x \in R$

from the primal problem :

$max_{f(.)} \int_R (2||x|| f(x) + \beta \sqrt{f(x)})dx\;\;\; \text{s.t.}\;\;\;\int_R f(x) dx= 1 \;\;\;\text{and}\;\;\;f(x) \geq 0\;\;\;\forall x \in R$

where $f(.)$ belongs to the Banach space $L²$ over a compact set $R$ (a distribution function). Do you know how to construct a dual problem in case the objective and constraints of the primal include integrals. It was said that standard techniques of vector space optimization could be used to approach the function $f(.)$, but this is maybe not obvious. I could not pinpoint the starting point.

I am reading a paper where the author derives the following Lagrangian dual problem :

$\min_v \int_R \frac{1}{4} \frac{\beta^2}{v-2\|x\|}dx+v\;\;\;\text{s.t.}\;\;\;v\geq 2\|x\|\;\;\;\forall x \in R$

from the primal problem :

$\max_{f(.)} \int_R (2\|x\| f(x) + \beta \sqrt{f(x)})dx\;\;\; \text{s.t.}\;\;\;\int_R f(x) dx= 1 \;\;\;\text{and}\;\;\;f(x) \geq 0\;\;\;\forall x \in R$

where $f(.)$ belongs to the Banach space $L^2$ over a compact set $R$ (a distribution function). Do you know how to construct a dual problem in case the objective and constraints of the primal include integrals. It was said that standard techniques of vector space optimization could be used to approach the function $f(.)$, but this is maybe not obvious. I could not pinpoint the starting point.