Abstract
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’skiĭ-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.
Highlights
IntroductionProblems of finding the zeros of monotone operators [ ], Proposition
Consider the following fixed point problem: Find x ∈ Fix(T) := x ∈ H : T x = x, ( . )Iiduka Fixed Point Theory and Applications (2016) 2016:77 where H stands for a real Hilbert space with inner product ·, · and its induced norm ·, T is a nonexpansive mapping from H into itself (i.e., T(x) – T(y) ≤ x – y (x, y ∈H)), and one assumes Fix(T) = ∅
This paper focuses on the Krasnosel’skiı-Mann algorithm, which has practical applications, such as analyses of dynamic systems governed by maximal monotone operators [ ] and nonsmooth convex variational signal recovery [ ], defined as follows: given the current iterate xn ∈ H and step size αn ∈ [, ], the iterate xn+ of the algorithm is xn+ := xn + αn T – xn
Summary
Problems of finding the zeros of monotone operators [ ], Proposition . There are useful algorithms for solving Problem ), such as the Krasnosel’skiı-Mann algorithm [ ], Subchapter . [ , ], and the hybrid method [ ] (Solodov and Svaiter [ ] proposed the hybrid method to solve problems of finding the zeros of monotone operators). This paper focuses on the Krasnosel’skiı-Mann algorithm, which has practical applications, such as analyses of dynamic systems governed by maximal monotone operators [ ] and nonsmooth convex variational signal recovery [ ], defined as follows: given the current iterate xn ∈ H and step size αn ∈ [ , ], the iterate xn+ of the algorithm is xn+ := xn + αn T (xn) – xn.
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