This paper considers a networked system consisting of an operator, which manages the system, and a finite number of subnetworks with all users, and studies the problem of minimizing the sum of the operator's and all users' objective functions over the intersection of the operator's and all users' constraint sets. When users in each subnetwork can communicate with each other, they can implement an incremental subgradient method that uses the transmitted information from their neighbor users. Since the operator can communicate with users in the subnetworks, it can implement a broadcast distributed algorithm that uses all available information in the subnetworks. We present an iterative method combining broadcast and incremental distributed optimization algorithms. Our method has faster convergence and a wider range of application compared with conventional distributed algorithms. We also prove that under certain assumptions our method converges to the solution to the problem in the sense of the strong topology of a Hilbert space. Moreover, we numerically compare our method with the conventional distributed algorithms in the case of a data storage system. The numerical results demonstrate the effectiveness and fast convergence of our method.

An unsharp masking technique (USMI is one of the image processing methods used in the computed radiography (CR) system. To further promote the performance of the USM, we developed the NEW-USM processing that can control the frequency enhancement characteristics flexibly and accommodate an extensive range of diagnostic targets. The NEW-USM and USM were performed on femur images acquired by computed radiography (model FCR9000; Fuji Medical Systems USA Inc, Stamford, CT), and the resulting images were compared. In the NEW-USM image, bone structures are enhanced as sharply as in the USM image, whereas the surrounding soft tissue structures, such as muscle are enhanced more strongly than in the USM image. Furthermore, the absence of the bone structure that may suggest pathological change is more obvious in the NEW-USM image. The newly developed NEW-USM can appropriately enhance diagnostic information over the whole range of image frequencies, thereby expanding utility of the USM. Copyright 9 1997 by W.B. Saunders Company KEY WORDS: computed radiography (CR), image enhancement, unsharp masking technique (USM), multi resolution analysis, spatial frequency response.

Variational inequality problems over fixed point sets of nonexpansive mappings include many practical problems in engineering and applied mathematics, and a number of iterative methods have been presented to solve them. In this paper, we discuss a variational inequality problem for a monotone, hemicontinuous operator over the fixed point set of a strongly nonexpansive mapping on a real Hilbert space. We then present an iterative algorithm, which uses the strongly nonexpansive mapping at each iteration, for solving the problem. We show that the algorithm potentially converges in the fixed point set faster than algorithms using firmly nonexpansive mappings. We also prove that, under certain assumptions, the algorithm with slowly diminishing step-size sequences converges to a solution to the problem in the sense of the weak topology of a Hilbert space. Numerical results demonstrate that the algorithm converges to a solution to a concrete variational inequality problem faster than the previous algorithm. MSC: 47H06; 47J20; 47J25

The existing machine learning algorithms for minimizing the convex function over a closed convex set suffer from slow convergence because their learning rates must be determined before running them. This paper proposes two machine learning algorithms incorporating the line search method, which automatically and algorithmically finds appropriate learning rates at run-time. One algorithm is based on the incremental subgradient algorithm, which sequentially and cyclically uses each of the parts of the objective function; the other is based on the parallel subgradient algorithm, which uses parts independently in parallel. These algorithms can be applied to constrained nonsmooth convex optimization problems appearing in tasks of learning support vector machines without adjusting the learning rates precisely. The proposed line search method can determine learning rates to satisfy weaker conditions than the ones used in the existing machine learning algorithms. This implies that the two algorithms are generalizations of the existing incremental and parallel subgradient algorithms for solving constrained nonsmooth convex optimization problems. We show that they generate sequences that converge to a solution of the constrained nonsmooth convex optimization problem under certain conditions. The main contribution of this paper is the provision of three kinds of experiment showing that the two algorithms can solve concrete experimental problems faster than the existing algorithms. First, we show that the proposed algorithms have performance advantages over the existing ones in solving a test problem. Second, we compare the proposed algorithms with a different algorithm Pegasos, which is designed to learn with a support vector machine efficiently, in terms of prediction accuracy, value of the objective function, and computational time. Finally, we use one of our algorithms to train a multilayer neural network and discuss its applicability to deep learning.

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