2015/09/18 by Hideaki Iiduka, Iiduka, Hideaki
Computer Science · Engineering · Mathematics · #47H10 #65K05 #90C25 #Advanced Optimization Algorithms Research #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1509.05605
openalex publication_date 2015/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper considers the fixed point problem for a nonexpansive mapping on a\nreal Hilbert space and proposes novel line search fixed point algorithms to\naccelerate the search. The termination conditions for the line search are based\non the well-known Wolfe conditions that are used to ensure the convergence and\nstability of unconstrained optimization algorithms. The directions to search\nfor fixed points are generated by using the ideas of the steepest descent\ndirection and conventional nonlinear conjugate gradient directions for\nunconstrained optimization. We perform convergence as well as convergence rate\nanalyses on the algorithms for solving the fixed point problem under certain\nassumptions. The main contribution of this paper is to make a concrete response\nto an issue of constrained smooth convex optimization; that is, whether or not\nwe can devise nonlinear conjugate gradient algorithms to solve constrained\nsmooth convex optimization problems. We show that the proposed fixed point\nalgorithms include ones with nonlinear conjugate gradient directions which can\nsolve constrained smooth convex optimization problems. To illustrate the\npracticality of the algorithms, we apply them to concrete constrained smooth\nconvex optimization problems, such as constrained quadratic programming\nproblems and generalized convex feasibility problems, and numerically compare\nthem with previous algorithms based on the Krasnosel'ski u i-Mann fixed point\nalgorithm. The results show that the proposed algorithms dramatically reduce\nthe running time and iterations needed to find optimal solutions to the\nconcrete optimization problems compared with the previous algorithms.\n