This paper proposes a constrained $ \ell_1 $ minimization method for estimating sparse precision matrices. The method, called CLIME, is shown to have desirable properties for estimating sparse precision matrices under both exponential-type and polynomial-type tail distributions. The CLIME estimator is derived by solving a constrained $ \ell_1 $ minimization problem, which ensures that the resulting estimator is symmetric and positive definite. Theoretical analysis shows that the CLIME estimator achieves a convergence rate of $ s\sqrt{\log p/n} $ in the spectral norm and $ \sqrt{\log p/n} $ in the elementwise $ \ell_\infty $ norm when the population distribution has either exponential-type or polynomial-type tails. The CLIME estimator also achieves faster convergence rates than $ \ell_1 $-MLE type estimators in the case of polynomial-type tails. The CLIME estimator is implemented via linear programming and is computationally efficient for high-dimensional data. It can be used for graphical model selection by adding a thresholding step. The method is tested on both simulated and real data, including a breast cancer dataset. The results show that the CLIME estimator performs favorably compared to existing methods in terms of both estimation accuracy and computational efficiency. The CLIME estimator is also shown to have good performance in terms of support recovery and sign consistency for the precision matrix. The method is compared to other methods such as the Graphical Lasso and SCAD, and is found to have better performance in terms of both estimation accuracy and computational efficiency. The CLIME estimator is also shown to have better performance in terms of classification accuracy when applied to the breast cancer dataset. The paper concludes that the CLIME estimator is a promising method for estimating sparse precision matrices in high-dimensional settings.This paper proposes a constrained $ \ell_1 $ minimization method for estimating sparse precision matrices. The method, called CLIME, is shown to have desirable properties for estimating sparse precision matrices under both exponential-type and polynomial-type tail distributions. The CLIME estimator is derived by solving a constrained $ \ell_1 $ minimization problem, which ensures that the resulting estimator is symmetric and positive definite. Theoretical analysis shows that the CLIME estimator achieves a convergence rate of $ s\sqrt{\log p/n} $ in the spectral norm and $ \sqrt{\log p/n} $ in the elementwise $ \ell_\infty $ norm when the population distribution has either exponential-type or polynomial-type tails. The CLIME estimator also achieves faster convergence rates than $ \ell_1 $-MLE type estimators in the case of polynomial-type tails. The CLIME estimator is implemented via linear programming and is computationally efficient for high-dimensional data. It can be used for graphical model selection by adding a thresholding step. The method is tested on both simulated and real data, including a breast cancer dataset. The results show that the CLIME estimator performs favorably compared to existing methods in terms of both estimation accuracy and computational efficiency. The CLIME estimator is also shown to have good performance in terms of support recovery and sign consistency for the precision matrix. The method is compared to other methods such as the Graphical Lasso and SCAD, and is found to have better performance in terms of both estimation accuracy and computational efficiency. The CLIME estimator is also shown to have better performance in terms of classification accuracy when applied to the breast cancer dataset. The paper concludes that the CLIME estimator is a promising method for estimating sparse precision matrices in high-dimensional settings.