Semidefinite Programming Approaches to Hankel Matrix Approximation and Completion via Primal–Dual Interior-Point Methods

Data completion techniques offer numerous advantages in various fields. However, completing large datasets that must satisfy specific criteria can be challenging, necessitating the use of approximative completion methods. The primary objective of this research paper is to showcase the substantial influence of the initial point selection on the completed matrix, emphasizing the distinctiveness of solutions corresponding to each initial point. However, in the case of using the same initial point, a unique solution is attainable although the employed completion methods may exhibit variations. Each method utilized for the completion of Hankel positive semidefinite matrices strives to achieve a unique solution, assuming feasibility. Nevertheless, the path to reaching this solution can differ among methods due to variations in the number of iterations and accuracy measures required for convergence. The paper will extensively explore theoretical aspects, algorithmic advancements, and empirical findings related to completing Hankel matrices through semidefinite programming (SDP) and combining SDP with second-order cone optimization (SOCP). The study will present numerical results to support the analyses conducted.