# ConvexOptimization

class boulderopal.noise_reconstruction.ConvexOptimization(power_density_lower_bound, power_density_upper_bound, regularization_hyperparameter)

Configuration for noise reconstruction with the convex optimization (CVX) method.

### Parameters

• power_density_lower_bound (float) – The lower bound for the reconstructed power spectral densities. It must be greater than or equal to 0.
• power_density_upper_bound (float) – The upper bound for the reconstructed power spectral densities. It must be greater than the power_density_lower_bound.
• regularization_hyperparameter (float) – The regularization hyperparameter $\lambda$

## Notes

The CVX method finds the estimation of the power spectral density (PSD) matrix ${\mathbf S}$

${\mathbf S}_{\mathrm{est}} = \mathrm{argmin}_{\textbf S} (\| F'{\mathbf S} - {\mathbf I} \|_2^2 + \lambda \| L_1 {\mathbf S} \|_2^2) ,$

where $F^\prime$ is the matrix of weighted filter functions and $| \bullet |_2$ denotes the Euclidean norm and $L_1$

\begin{align} L_1 = \begin{bmatrix} -1 & 1 & & \\ & \ddots & \ddots & \\ & & -1 & 1 \\ \end{bmatrix}_{(K - 1) \times K} . \end{align}

$\lambda$ is a positive regularization hyperparameter which determines the smoothness of ${\mathbf S}_{\mathrm{est}}$