(S+\gamma~t~I)/(1+\gamma),$$ where $I$ is the identity matrix, and $t=\sum_{i=1}^{p}S_{ii}/p$. After some algebra, conclude that for any $\gamma>0$: $$\begin{aligned}
\sigma_{min} (\hat{\Sigma}) & = & \sigma_{min} (S+ \gamma~t~I)/(1+\gamma) \\
& \geq & \frac{\gamma~t}{1+\gamma}>0\end{aligned}$$ This means that the eigenvalues variance penalty improves $S$ to a positive definite estimator $\hat{\Sigma}$. However the estimator (2.3) is well-conditioned but need not be sparse. Sparsity can be achieved by imposing $\ell_1$ penalty on the entries of covariance matrix. Simulations have shown that, in general the minimizer of (2.1) is not positive definite for all values of $\lambda >0$ and $\gamma >0$. Here onwards we focus on correlation matrix estimation, and later generalize the method for covariance matrix estimation.\
To achieve both well-conditioned and sparse positive definite estimator we optimize the following objective function in $R$ over specific region of values of $(\lambda, \gamma)$ which depends upon sample correlation matrix $K$ and $\lambda,\gamma$. Here the condition $tr(\Sigma)=tr(S)$ reduces to $tr(R)=p$, and $t=1$. Consider the following optimization problem: $$\hat{R}_K=\operatorname*{arg\,min}_{R=R^T,tr(R)=p|(\lambda, \gamma) \in \hat{\mathscr{S}}^{K}_1}~~\Big[  ||R-K||^2_F +  \lambda \|R^-\|_1  + \gamma\sum_{i=1}^{p} \big \{\sigma_i(R)-\bar{\sigma}_{R} \big \}^2\Big],$$ where

$$\begin{aligned}
\hat{\mathscr{S}}^{K}_1& = \Big \{(\lambda,\gamma): \lambda, \gamma >0, \lambda \asymp \gamma \asymp \sqrt{\frac{log~ p}{n}}, \forall \epsilon >0,\sigma_{min}\{(K+\gamma I)-\frac{\lambda}{2}*sign(K+\gamma I)\}>\epsilon \Big \},\end{aligned}$$

and $\bar{\sigma}_{R}$ is mean of 