ON ACCELERATION OF THE KRASNOSEL'SKII-MANN FIXED POINT ALGORITHM BASED ON CONJUGATE GRADIENT METHOD FOR SMOOTH OPTIMIZATION

This paper considers the problem of finding a fixed point of a nonexpansive mapping on a real Hilbert space and proposes a novel algorithm to accelerate the Krasnosel'skii-Mann algorithm. To this goal, we first consider an unconstrained smooth convex minimization problem, which is an example of a fixed point problem, and show that the Krasnosel'skii-Mann algorithm to solve the minimization problem is based on the steepest descent method. Next, we focus on conjugate gradient methods, which are popular acceleration methods of the steepest descent method, and devise an algorithm blending the conjugate gradient methods with the Krasnosel'skii-Mann algorithm. We prove that, under realistic assumptions, our algorithm converges to a fixed point of a nonexpansive mapping in the sense of the weak topology of a Hilbert space. We perform convergence rate analysis on our algorithm. We numerically compare our algorithm with the Krasnosel'skii-Mann algorithm and show that it reduces the running time and iterations needed to find a fixed point compared with that algorithm.