Study On Q-stable Point Of Zero-norm Constrained Optimization Problem

Zero-norm constrained optimization is widely used in portfolio problems,machine learning,signal and image processing,pattern recognition,regression analysis and many other fields.Because of the combinativity of zero-norm functions,the zero-norm constrained optimization problem is generally NP-hard,and the general continuous optimization method is not suitable to deal with this problem.In order to overcome the difficulties caused by the combinatorial properties of zero-norm functions,this paper transform the NP-hard problems into constrained optimization problems with semi-complementary constraint sets,i.e.MPCC problems,by means of the variational characterizations of zero-norm functions.In this thesis,the stability point theory of MPCC problem is studied.The description of tangent cone and normal cone of feasible set of constrained optimization problem plays an important role in describing the optimality condition of the optimization problem.On this basis,in chapter 3,the exact expressions of the tangent cone,regular normal cone and limit normal cone of the set of semi-complementary constraints are given.With the help of semi-complementary set tangent cone and normal cone characterizations,the forth chapter of the thesis proposes.The definition of Q stable point of MPCC problem is given.Then,the relation between Q stable point and M,S stable point is studied.In particular,the relation between the stable points of MPCC and NZCP problem is discussed when the zero-norm constraint optimization problem NZCP has only zero-norm constraint.