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Solving l1 regularized optimization problems is common in the fields of computational biology, signal processing and machine learning. Such l1 regularization is utilized to find sparse minimizers of convex functions. A well-known example is the LASSO problem, where the l1 norm regularizes a quadratic function. A multilevel framework is presented for solving such l1 regularized sparse optimization problems efficiently. We take advantage of the expected sparseness of the solution, and create a hierarchy of problems of similar type, which is traversed in order to accelerate the optimization process. This framework is applied for solving the sparse inverse covariance estimation problem, where the inverse of an unknown covariance matrix of a multivariate normal distribution is estimated, under the assumption that it is sparse. To this end, an l1 regularized log-determinant optimization problem needs to be solved. This task is challenging especially for large-scale data sets, due to time and memory limitations. Numerical experiments demonstrate the efficiency of the multilevel framework for both medium and large scale instances of this problem.