Alternative gradient algorithms for computing the nearest correlation matrix

The correlation matrix has a wide range of applications in finance and risk management. However, due to the constraints of practical operations, the correlation matrix cannot satisfy the positive semidefinite property in most cases. In this paper, an elementwisely alternative gradient algorithm and a columnwisely alternative gradient algorithm are presented to compute the nearest correlation matrix that satisfies the semidefinite property for a given set of constraints. The convergence properties and the implementation of these two algorithms are discussed. Numerical experiments show that the proposed methods are efficient. Furthermore, the columnwisely alternative gradient algorithm outperforms other algorithms in terms of the number of iterations and the objective value of the cost function.