On Convergence of the Maximum Block Improvement Method

The so-called MBI (maximum block improvement) method is a greedy type approach to solve optimization problems where the decision variables can be grouped into a finite number of blocks. Assuming that optimizing over one block of variables while fixing all others is relatively easy, the MBI method then advocates to update the block of variables corresponding to the maximally improving block at each iteration, which is arguably a most natural and simple process to tackle block-structured problems with great potentials for engineering applications. It is therefore imperative to fully understand its convergence properties. In this paper we establish global and local linear convergence results for this method. The global convergence is established under the Lojasiewicz inequality assumption, while the local analysis invokes second-order assumptions. We study in particular the tensor optimization model with spherical constraints. Conditions for linear convergence of the famous power method for computing the maximum eigenvalue of a matrix follow in this framework as a special case. The condition is interpreted in various other forms for the rank-one tensor optimization model under spherical constraints. Numerical experiments are shown to support the convergence property of the MBI method for the randomly generated instances.