the eigenvalues of $R$. For instance when $K$ is diagonal matrix, the set $\hat{\mathscr{S}}^{K}_1$ is given by:

$\hat{\mathscr{S}}^{K}_1 = \Big \{(\lambda,\gamma): \lambda, \gamma >0, \lambda \asymp \gamma \asymp \sqrt{\frac{log~ p}{n}}, \forall \epsilon >0,\lambda <2(\gamma-\epsilon) \Big \}$.

The minimization in (2.4) over $R$ is for fixed $ (\lambda, \gamma) \in \hat{\mathscr{S}}^{K}_1$. The proposed estimator of covariance matrix (based on regularized correlation matrix estimator $\hat{R}_K$) is given by $\hat{\Sigma}_K=({S^+})^{1/2}\hat{R}_K({S^+})^{1/2}$, where $S^+$ is the diagonal matrix of the diagonal elements of $S$. Furthermore Lemmas 3.1 and 3.2, respectively show that the objective function (2.4) is convex and estimator given in (2.4) is positive definite.

Our Contribution
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The main contributions are the following:\
**i)** The proposed estimators are both sparse and well-conditioned simultaneously. This approach allows to take advantage of a prior structure if known on the eigenvalues of the true covariance and the inverse covariance matrices.\
**ii)** We establish theoretical consistency of proposed estimators in both operator and Frobenius norm. The proposed JPEN estimators achieves the minimax rate of convergence in operator norm for the underlying class of sparse and well-conditioned covariance and inverse covariance matrices and therefore is optimal.\
**iii)** The proposed algorithm is very fast, efficient and easily scalable to 