Semi-definite programming for the nearest circulant semi-definite matrix problem

Positive semi-definite circulant matrices arise in many important applications. The problem arises in various applications where the data collected in a matrix do not maintain the specified structure as is expected in the original system. The task is to retrieve useful information while maintaining the underlying physical feasibility often necessitates search for a good structured approximation of the data matrix. This paper construct structured circulant positive semi-definite matrix that is nearest to a given data matrix. The problem is converted into a semi-definite programming problem as well as a problem comprising a semi-defined program and second-order cone problem. The duality and optimality conditions are obtained and the primal-dual algorithm is outlined. Some of the numerical issues involved will be addressed including unsymmetrical of the problem. Computational results are presented.