# Dual problem with integrals

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.

## 1 Answer

$$\newcommand\R{\mathbb R}$$ A convenient way to derive the dual problem from a primal one is by using the minimax duality for the Lagrangian, which is given here by the formula $$L(f,v):=\int_R\big[2|x|f(x)+b\sqrt{f(x)}\big]\,dx-v\Big(\int_Rf(x)\,dx-1\Big),$$ where $$|x|:=\|x\|$$ and $$b:=\beta$$. Clearly, $$\sup_{f\ge0}\inf_{v\in\R} L(f,v) =\sup\Big\{\int_R\big[2|x|f(x)+b\sqrt{f(x)}\big]\,dx\colon f\ge0,\int_R f(x)\,dx=1\Big\},$$ which is value of the primal problem.

The value of the dual problem is $$\inf_{v\in\R}\sup_{f\ge0} L(f,v) =\inf_{v\in\R}\Big(v+\sup\Big\{\int_R\big[2|x|f(x)+b\sqrt{f(x)}-vf(x)\big]\,dx\colon f\ge0\Big\}\Big) =\inf_{v\in\R}\Big(v+\int_R s(|x|,v)\,dx\Big),$$ where $$s(a,v):=\sup\{2at+b\sqrt t-vt\colon t\ge0\}.$$ For any real $$a>0$$, it is easy to see that $$s(a,v)=\frac{b^2}{4 (v-2 a)}$$ if $$b>0$$ and $$v>2a$$, $$s(a,v)=0$$ if $$b\le0$$ and $$v\ge2a$$, and $$s(a,v)=\infty$$ otherwise; in particular, $$s(a,v)=\infty$$ if $$v<2a$$. Thus, with $$|R|:=\max\{|x|\colon x\in R\}$$, the value of the dual problem is $$\inf_{v>2|R|}\Big(v+\int_R \frac{\max(0,b)^2}{4 (v-2|x|)}\,dx\Big).$$

• The equality in the interchanging of $sup_{f}$ and $\int_R$ comes from the continuity and the measurability of the function $f(.)$, right. – Omar Delgado Sep 19 '20 at 19:52
• @OmarR. : The inrterchange of $\sup_f$ and $\int_R$ is possible because, for $v>2a$, the maximizer of $2at+b\sqrt t-vt$ in $t\ge0$ is $\max(0,b)^2/(4(v-2a)^2)$, which is a measurable function of $a$ ($a$ here representing $|x|$). – Iosif Pinelis Sep 20 '20 at 1:06
• Thank you very much. – Omar Delgado Sep 20 '20 at 10:17