The Penalized Analytic Center Estimator

In a linear regression model, the Dantzig selector (Candès and Tao, 2007) minimizes the L 1 norm of the regression coefficients subject to a bound λ on the L ∞ norm of the covariances between the predictors and the residuals; the resulting estimator is the solution of a linear program, which may be nonunique or unstable. We propose a regularized alternative to the Dantzig selector. These estimators (which depend on λ and an additional tuning parameter r ) minimize objective functions that are the sum of the L 1 norm of the regression coefficients plus r times the logarithmic potential function of the Dantzig selector constraints, and can be viewed as penalized analytic centers of the latter constraints. The tuning parameter r controls the smoothness of the estimators as functions of λ and, when λ is sufficiently large, the estimators depend approximately on r and λ via r / λ -super-2.