Computing the Nearest Correlation Matrix using Difference Map Algorithm

The difference map algorithm (DMA) is originally designed to find the global optimal solution to nonconvex problems. The main feature of DMA is that it can avoid the stagnation, (which always occurs when applying the alternating projection method (APM) on nonconvex problems so that APM is trapped at local minimized point and fail to find the global optimal solution.) and converge to the global solution directly. We show that DMA is not only a better algorithm to solve nonconvex problems but also is better to solve convex problems. We using the nearest correlation matrix to show this statement. The nearest correlation matrix (NCM) problem is to find the nearest point to a given symmetric matrix A in the intersection of positive semi-definite cone and the unit diagonal space. This is a very typical example of convex problems. Currently, three popular ways to solve this problem are: the alternating projection method ; (quadratic) Newton’s method ; and the primal-dual interiorexterior-point approach. The least expensive of these methods is the alternating projection method. However, this method can only achieve a linear rate of convergence and may cause a huge computational cost when we are looking for a highly accurate solution. Because of the similarity of DMA and APM, we are going to compare the best results obtained by both algorithms and the computational cost with different desired accuracies of the final solution. Numerical experiments show that DMA with β value equals to 1 can obtain the best solution and it is identical to the one obtained by APM. However, when A is large sized and ill-conditioned, or we are seeking a highly accurate solution, DMA can find the optimal solution with less computational cost.