Dykstra’s algorithm for constrained least-squares doubly symmetric matrix problems

In this work we apply Dykstra’s alternating projection algorithm for minimizing ‖AX−B‖ where ‖⋅‖ is the Frobenius norm and A∈Rm×n, B∈Rm×n and X∈Rn×n are doubly symmetric positive definite matrices with entries within prescribed intervals. We first solve the constrained least-squares matrix problem by using the special structure properties of doubly symmetric matrices, and then use the singular value decomposition to transform the original problem into a simpler one that fits nicely with the algorithm originally developed by [R. Escalante, M. Raydan, Dykstra’s algorithm for a constrained least-squares matrix problem, Numer. Linear Algebra Appl. 3 (1996) 459–471].