# Constraint optimization with Calculus of Variations. How to handle positive function constraint?

the I am attempting to maximize the functional $F[f]$ with a constrain that $f$ has to be non-negative and some other integral constraints. More, specifically, \begin{align*} &\max F[f]\\ &\text{ s.t. } f(x) \ge 0, \forall x\\ &\text{ and} \int_{-\infty} ^\infty f(x)dx= c \end{align*} where $c$ is some constant.

I decide to approach this with the method of Lagrangian multipliers (if you know a better method that would be great too). My Lagrangian equation is the following:

\begin{align*} L[f] =F[f]- \lambda_0(f(x)-0) -\lambda_1 \left(\int_{-\infty} ^\infty f(x)dx - c\right) \end{align*}

The problem now is that I have to take the variational derivative with respect to $f$. My main question is, how do I handle the first constrain, since it is not in the form of an integral?

Variational derivative of $f(x)$ with respect to $f(x')$(need a new dummy variable) is $\delta(x-x')$ which introduces a new dummy variable $x'$. So, we get \begin{align*} \frac{L[f]}{d f(x')} =\frac{F[f]}{d f(x')}-\lambda_0\delta(x-x') -\lambda_1 \end{align*}

My question is how do people generally deal with the constraint that the function has to be non-negative when using calculus of variation? Is there a work around this? Or introduction of a dummy variable is inevitable?

I would be grateful for any help. Thank you.

• I don't remember the theory, but here's the intuition: If you only enforced nonnegativity at a finite number of points $x_0, x_1, x_2, \ldots$, your Lagrangian would be $L[f] = F[f]-\lambda_0(f(x_0)-0)-\lambda_1(f(x_1)-0)-\lambda_2(f(x_2)-0)-\cdots$ with $\lambda_i\ge0$. But nonnegativity at all points is an infinite number of constraints. So you may consider $\lambda$ a function of $x$, and write the Lagrangian as $$L[f] = F[f]-\int\lambda(x)(f(x)-0)\,\mathrm dx,$$ with $\lambda(x)\ge0$ for all $x$. – Rahul Jan 30 '15 at 23:50
• @Rahul Thank you very much. Do you know of any accessible books on this subject? – Boby Feb 2 '15 at 3:33
• I don't know. I've only read the textbook by Gelfand and Fomin. It wasn't too forbidding, as I recall. – Rahul Feb 2 '15 at 4:10
• Ayyyyyyy. Thanks – Boby Feb 3 '15 at 5:31