This paper investigates the [Formula: see text]-regularization problems, which has a broad applications in compressive sensing, variable selection problems and sparse least squares fitting for high dimensional data. We derive the exact lower bounds for the absolute value of nonzero entries in each global optimal solution of the model, which clearly demonstrates the relation between the sparsity of the optimum solution and the choice of the regularization parameter and norm. We also establish the necessary condition for global optimum solutions of [Formula: see text]-regularization problems, i.e., the global optimum solutions are fixed points of a vector thresholding operator. In addition, by selecting parameters carefully, a global minimizer which will have certain desired sparsity can be obtained. Finally, an iterative thresholding algorithm is designed for solving the [Formula: see text]-regularization problems, and any accumulation point of the sequence generated by the designed algorithm is convergent to a fixed point of the vector thresholding operator.
Keywords: Fixed point; Global optimum solution; Iterative thresholding algorithm; Optimality condition; [Formula: see text]-regularization problems.