Alternating projection method for a class of tensor equations

This paper considers how to solve a class of tensor equations arising from the unified definition of tensor-vector products. Of special interest is the order-3 tensor equation whose solutions are the intersection of a group of quadrics from a geometric point of view. Inspired by the method of alternating projections for set intersection problems, we develop a hybrid alternating projection algorithm for solving order-3 tensor equations. The local linear convergence of the alternating projection method is established under suitable conditions. Some numerical experiments are conducted to evaluate the effect of the proposed algorithm.