| The optimization problems with cardinality constraint have a wide range of applications in the fields of investment portfolio,signal processing,compressed sensing and image processing,and become one of the most challenging optimization problems.Because of the existence of cardinality constraint,the optimization problem with cardinality constraint is NP-hard.Differ-ent approximation algorithms can be designed by using different approximation functions of7)0-norm,and the quality of solutions for optimization problems with cardinality constraint is closely related to the selection of approximation functions of7)0-norm.This paper aims to study the successive convex optimization（SCO）algorithm and its numerical implementation for op-timization problems with cardinality constraint under the non-convex approximation functions of7)0-norm.The following are the main results of this paper:（1）We study the SCO algorithm for convex optimization problems with cardinality con-straint based on the nonlinear DC（Difference of Convex functions,DC）approximation function and its global convergence.Firstly,a nonlinear DC approximation function with7)0-norm is pre-sented,and a non-convex approximation problem of the original problem is obtained.Secondly,a series of convex approximation subproblems are constructed by linearizing the concave term of the DC function,and then the SCO algorithm for solving non-convex approximation prob-lem is proposed.It is proved that the SCO algorithm converges to the KKT point of the DC approximation problem.Numerical experiments show that the SCO algorithm based on nonlin-ear DC approximation function proposed in this paper can effectively find the sparse solutions of the convex optimization problems with cardinality constraint,and has certain advantages in the quality of solutions.（2）We study the SCO algorithm for DC optimization problems with cardinality constraint and conduct its numerical implementation.Firstly,using a general non-convex approximation function to approximate the7)0-norm,we obtain a DC approximation problem of the original problem.Then,by linearizing the concave term in the objective and constraint functions of the DC approximation problem,we obtain a convex approximation problem.Secondly,based on the convex approximation problem,an SCO algorithm for solving the DC approximation problem is proposed,and it is proved that the SCO algorithm converges to the KKT point of the DC approximation problem.Numerical experiments show that the proposed SCO algorithm can effectively find sparse solutions for DC optimization problems with cardinality constraint. |
