Sparsity is crucial for high-dimensional statistical modeling. On one hand, dimensionality reduction can reduce the variability of estimation and thus provide reliable predictive power. On the other hand, the selected sub-model can discover and emphasize the underlying dependencies, which is useful for objective interpretation. Many variable selection methods have been proposed in literatures. For a prominent example, Least Absolute Shrinkage and Selection Operator (lasso) in linear regression context has been extensively explored. This paper discusses a class of scaled mixture of Gaussian models from both a penalized likelihood and a Bayesian regression point of view. We propose an Majorize-Minimize (MM) algorithm to find the Maximum A Posteriori (MAP) estimator, where the EM algorithm can be stuck at local optimum for some members in this class. Simulation studies show the outperformance of proposed algorithm in nonstochastic design variable selection scenario. The proposed algorithm is applied to a real large-scale E.coli data set with known bona fide interactions for constructing sparse gene regulatory networks. We show that our regression networks with a properly chosen prior can perform comparably to state-of-the-art regulatory network construction algorithms.