  This paper considers the fixed point problem for a nonexpansive mapping on a
real Hilbert space and proposes novel line search fixed point algorithms to
accelerate the search. The termination conditions for the line search are based
on the well-known Wolfe conditions that are used to ensure the convergence and
stability of unconstrained optimization algorithms. The directions to search
for fixed points are generated by using the ideas of the steepest descent
direction and conventional nonlinear conjugate gradient directions for
unconstrained optimization. We perform convergence as well as convergence rate
analyses on the algorithms for solving the fixed point problem under certain
assumptions. The main contribution of this paper is to make a concrete response
to an issue of constrained smooth convex optimization; that is, whether or not
we can devise nonlinear conjugate gradient algorithms to solve constrained
smooth convex optimization problems. We show that the proposed fixed point
algorithms include ones with nonlinear conjugate gradient directions which can
solve constrained smooth convex optimization problems. To illustrate the
practicality of the algorithms, we apply them to concrete constrained smooth
convex optimization problems, such as constrained quadratic programming
problems and generalized convex feasibility problems, and numerically compare
them with previous algorithms based on the Krasnosel'ski\u\i-Mann fixed point
algorithm. The results show that the proposed algorithms dramatically reduce
the running time and iterations needed to find optimal solutions to the
concrete optimization problems compared with the previous algorithms.
