The Numerical Algorithm For Solving The Nearest Correlation Matrix Problem

The nearest correlation matrix problem is to find a correlation matrix which is nearest to a given symmetric matrix in the Frobenius norm, and it includes many types such as the none-weighted, the W-weighted, the H-weighted and the Q-weighted problem, etc. The thesis is mainly concerned with theories and numerical methods of the first three types.The research status and advances of the nearest correlation matrix problem are introduced firstly, and then the working assumptions are put forward in this thesis.Meanwhile, some basic knowledge and related optimization algorithms are given in to facilitate the next research.In chapter 2, theories and numerical methods of the none-weighted problem are mainly analyzed, which is a special form of the W-weighted problem. Then The relationship between the none-weighted problem and its corresponding dual problem is analyzed, namely the solution of primal problem can be expressed by the solution of its corresponding dual problem. As the dual problem is equivalent to a semismooth equations, based on the Newton method solving the system of semismooth equations,the semismooth equations is first modified by using regularized strategy, and then a regularization Newton method is prosposed to the modified semismooth equations,which ensures the computed search direction to be a descent direction of objective function and overcomes the inherent defects of semismooth Newton method effectively. In order to improve the efficiency of the regularization Newton method,namely inner iteration, a preconditioned strategy in the conjugate gradient method for semismooth equations and the choice of inner iteration control variable are analyzed,then a modified regularization Newton method is proposed to solve the modified semismooth equations, and its global convergence and quadratic convergence rate are established. Finally, correlation processing to the computed solution obtained by the modified regularization Newton method is introduced to make the final calculated solution closer to correlation matrix, and the corresponding error estimate with the optimal solution is analyzed. As the W-weighted problem can be transformed into none-weighted problem, so the above work about none-weighted problem is simply extended to the W-weighted problem.In chapter 3, the theories and numerical methods of the H-weighted problem are analyzed. First of all, the constraint nondegeneracy property, strong second ordersufficient condition and the calculating formula of generalized Jacobi are given. Then the Newton-conjugate gradient method(Newton-CG method) for solving H-weighted problem is modified. Forcing terms in inner iteration will be redefined by using the first and second order information of the objective function at each iteration point,which can balance the iteration computation between internal and external interation and improve the efficiency of the algorithm. The modified Newton-CG method still has a quadratic convergence by the simple analysis.Finally, numerical experiments are reported in the thesis. The numerical results show that the proposed methods are practical.