A Difference Convex Method For Miniming A Quartic Form Over The Sphere

DC methods can be applied to solve the quartic minimization problem over the sphere.In this paper,a DC algorithm is proposed,and its convergence analysis is presented.Some preliminary numerical results show that the proposed DC algorithm has some advantages over the method built-in MATLAB and power method on global solutions.Moreover,the DC algorithm is less sensitive on the initial points,and global optimality is exhibited frequently.Particularly,the accuracy of the DC method is better for large scale problems.Firstly,we introduces the current research status of tensor decomposition and its approximation,the best rank one approximation of tensors,and the fourth-order minimization problem.Secondly,the concept of matrix subdifferential is introduced,and the general form of subdifferential of matrix nuclear norm and matrix spectral norm is explained.Furthermore,the DC planning problem is introduced,the general form of the DC planning problem and its classic DC algorithm and high-order power method are given,and the convergence of the algorithm is introduced.Then,the DC algorithm is used to solve the quartic minimization problem under the spherical constraint,the algorithm convergence analysis is given,and the corresponding numerical experiments are carried out.Finally,the application of the quadratic minimization problem to the stable set problem of graph theory is introduced,and the calculation results of specific examples and random examples are given.