Research On The Least Squares Problem Of Linear Matrix Inequality With Closed Convex Cone Constraint

The problem that finds the structural matrices that satisfy the given linear matrix inequality and constraint conditions is called the linear constraint matrix inequality problem.This problem which arises from many scientific and engineering fields such as matrix optimization,inverse problems of radiotherapy,and image reconstruction.But in many practical applications,the given data of the linear constraint matrix inequality problem usually has measurement or observation errors.So the corresponding linear constraint matrix inequality problem may be incosistent,which means that there is no matrix that satisfies the corresponding linear matrix inequality on the given constraint set.Consequently,it is necessary to establish the least square model of the incosistent linear constrained matrix inequality problem,and to investigate the corresponding theories and algorithmsSimilar to the typical linear least squares problem,the objective function of the least squares problem of linear constraint matrix inequality is convex and the gradient of this objective function is global Lipschitz continuous.So the constrained least squares solution of this problem always exists so long as the corresponding constrained set is a closed convex.However,the second-order Jacobi matrix of the objective function of this type of problem does not exist,and the unknown matrix also has a special structure such as symmetric or bisymmetric in many practical applications,image reconstruction for example.Hence,it is not feasible to directly apply the numerical methods which solve the typical linear least squares problem to this kind of special matrix least squares problem.In addition,the current research results of this least squares problem mainly include the theories and algorithms about the unconstrained least squares problem or the problem with subspace constraints.So it is necessary to further study the linear matrix inequality problem on given closed convex cone as well as its least squares problemThis paper systematically investigates the theories of the linear matrix inequality problems and its least squares problems with general closed convex cone constraints,and attempts to give a general algorithm framework for solving those problems.To this end,the least squares solution of this kind of problem has been firstly characterised in this paper in view of the best approximation theorem and the polar decomposition theorem in Hilbert space.Furthermore,the iterative algorithm framework based on the ideas of existing algorithms is proposed to solve this type of least squares problem,and the convergence theory of the algorithm framework is also obtained in this paper.Moreover,the iterative algorithm framework has been applied to solve a kind of linear matrix inequality problem with non-negative symmetric structural constraints.The key of this algorithm is to solve a classical constrained least square subproblems.In this paper,the spectral projection gradient method for solving the minimum problem of smooth function under convex constraints is used to solve this subproblem,and then a computable algorithm has been obtained.Finally,several numerical examples have been listed to verify the correctness of theoretical results and the effectiveness of given numerical algorithms.