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The two-cardinality sparse constrained optimization problems include sparse optimization problems and constrained sparse optimization problems in many fields, such as signal processing, image processing, securities investment etc. A smoothing penalty function method and a smoothing objective penalty function method are studied for two-cardinality sparse constrained optimization problems respectively. Some error estimations are proved for the smoothing penalty function and the smoothing objective penalty function. Based on the smoothing penalty function and smoothing objective penalty function, two algorithms are designed to solve two-cardinality sparse constrained optimization problems and their convergence is proved respectively. Numerical results show that the two algorithms have the similar effectiveness in finding out an approximate solution for two-cardinality sparse constrained optimization problems.

The smoothing objective penalty function method for two-cardinality sparse constrained optimization problems