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

DOI: https://doi.org/10.37193/CJM.2021.01.02

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.