An Approximate Inexact Accelerated Proximal Gradient Method For The Minimization Problem Of A Class Of Maximum Eigenvalue Functions

Nonsmooth optimization is an important branch of operations research. The optimization of eigenvalue in the nonsmooth optimization is a research method which is used widely in engeering, physics and statistics fields. The sum of two nonsmooth functions are nonsmooth too. In this paper, an approximate inexact accelerated proximal gradient (AIAPG) method is discussed to solve the minimization problem of a class of maximum eigenvalue functions. The following minimization problem of a class of maximum eigenvalue functions is considered:(P) min{λmax(X)+g(X):X∈Sn}, where the function g(X) is a proper, lower semi-continuous, nonsmooth convex function, dom(g)={X∈Sn:g(X)<∞} is closed. The main content in this paper is describes as follows:In the second chapter of this paper, avoiding the complexity of the minimization of the sum of two nonsmooth functions, we consider smooth approximation to the maximum eigenvalue function such that the function is a proper, lower semi-continuous, nonsmooth convex function. So we get the equivalent problem. This process lays the foundation for the next chapter. In the third chapter, we give approximate exact accelerated proximal gradient (AAPG) method for solving the minimization problem of a class of maximum eigenvalue functions. Considering the the complexity in computation, we give our AIAPG method. The inexactly property in the AIAPG method means that each iteration in solving the subproblem is approximately. We prove the global convergence in this inexactly property. In the fifth chapter, we apply our AIAPG method to solve the minimization problem of maximum eigenvalue functions with linear constrains. In this paper’s final chapter, we analyze the situation which appears in our real life, such as the matrix X does not belong to feasible region C, where C={X∈Sn:A (X)=b,X(?)O}. We prove the global convergence.