A revised MRMIL Riemannian conjugate gradient method with simplified global convergence properties

Abstract

In this work, we propose an effective coefficient for the conjugate gradient (CG) method. First, we present the coefficient for Euclidean optimization, explaining its motivation, and then extend it to Riemannian optimization. We analyze the convergence of the CG method generated by this coefficient in the context of Riemannian optimization, ensuring that the generated search direction satisfies the sufficient descent property. This property ensures that the sequence generated converges to a minimizer of the underlying function. We test the effectiveness of the proposed coefficient numerically on various Riemannian optimization problems, demonstrating favorable performance compared to existing Riemannian CG methods and other coefficients of similar class. These results also extend to Euclidean optimization, where such findings have not yet been established.