Consider the minimization problem described this paper. Let $f_{\lambda}$ be the minimizer. As a part of extending my work, I am able to show the following facts

$$\lim_\limits{\lambda \to 0}\|f_{\lambda}\|_{L^2} = 0$$ and $$\lim_\limits{\lambda \to \infty}\|f_{\lambda}\|_{L^2} = 0$$

My problem now is (as I would like to extend my work), find $\lambda \in (0,\infty)$ for which $\|f_{\lambda}\|_{L^2}$ is maximum. Appreciate your suggestions to solve this problem.

The minimization problem from the linked paper is given below for the self containment of the post. If given that $k>\frac{m}{2}$, the paper proves that there is a unique minimizer for the functional $C(f)$ in the set $S$.

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It is given that $k>\frac{m}{2}$

My progress

Partial Progress :

Progress : I am able to derive the corresponding PDE equations for the problem.

Let $f(\lambda,x) = f_{\lambda}(x)$. Then

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The first equation corresponds to maximizing $\|f_{\lambda}\|$, while the second PDE is for the minimization problem associated with the parameter $\lambda$.

The second equation (minimization problem), given any $\lambda$, I can solve for $f(\lambda,.)$ either using linear algebra or steepest descent algorithm, which I have described in my article. Now I need to use this solution and the first equation to obtain $\lambda$, which is a problem I am facing.

Trying to solve using linear algebra, by formulating the discrete version of the problem using Fourier series coefficients and Plancheral theorem, I get stuck at the matrix problem described here.

More Partial Progress

An Iterative algorithm which is a modified steepest descent.

  1. Initialize $f$.

  2. Assuming some $\lambda$ and assuming gradient of $C_{\lambda}(f)$ wrt $f$ be $\nabla_f C_{\lambda}(f)$, and if we were to update $f$ with this gradient as in we do in steepest descent, it would be $f^u_\lambda = f - \delta \nabla_f C_{\lambda}(f)$, where $\delta$ is a constant learning rate. Now set $\frac{\partial\|f^u_\lambda\|}{\partial \lambda} = 0$ and solve for $\lambda$. Let the root be $\lambda_0$.

  3. Update $f = f^u_{\lambda_0}$. (update $f$ as in steepest descent, but using $\lambda$ value as $\lambda_0$ which was computed in step 2.)

  4. check some convergence criterion and if not met, go to step 2.

I have implemented this numerically and it converges as desired. Need to work on the proof.

PS : This was first posted on MO by me, 3 months back. Link and cross posted on math.SE here.


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