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L2 and L1 constrained regression methods, such as ridge regression and Lasso, have been generally known for their fitting ability. Recently, L0- constrained classifications have been used for feature selection and classifier construction. This paper proposes an L0 constrained regression method, which aims to minimize both the epsilon-insensitive fitting errors and L0 constraints on regression coefficients. Our L0-constrained regression can be efficiently approximated by successive linearization algorithm, and shows the favorable properties of selecting a compact set of fitting coefficients and tolerating small fitting errors. To make our L0 constrained regression generally applicable, the extension to nonlinear regression is also addressed in this paper.