Hideaki Iiduka

A Use of Conjugate Gradient Direction for the Convex Optimization Problem over the Fixed Point Set of a Nonexpansive Mapping

In this paper, we discuss the convex optimization problem over the fixed point set of a nonexpansive mapping. The main objective of the paper is to accelerate the hybrid steepest descent method for the problem. To this goal, we present a new iterative scheme that utilizes the conjugate gradient direction. Its convergence to the solution is guaranteed under certain assumptions. In order to demonstrate the effectiveness, performance, and convergence of our proposed algorithm, we present numerical comparisons of the algorithm with the existing algorithm.

Fixed point optimization algorithm and its application to power control in CDMA data networks

We discuss the variational inequality problem for a continuous operator over the fixed point set of a nonexpansive mapping. One application of this problem is a power control for a direct-sequence code-division multiple-access data network. For such a power control, each user terminal has to be able to quickly transmit at an ideal power level such that it can get a sufficient signal-to-interference-plus-noise ratio and achieve the required quality of service. Iterative algorithms to solve this problem should not involve auxiliary optimization problems and complicated computations. To ensure this, we devise a fixed point optimization algorithm for the variational inequality problem and perform a convergence analysis on it. We give numerical examples of the algorithm as a power control.

Iterative Algorithm for Solving Triple-Hierarchical Constrained Optimization Problem

Many practical problems such as signal processing and network resource allocation are formulated as the monotone variational inequality over the fixed point set of a nonexpansive mapping, and iterative algorithms to solve these problems have been proposed. This paper discusses a monotone variational inequality with variational inequality constraint over the fixed point set of a nonexpansive mapping, which is called the triple-hierarchical constrained optimization problem, and presents an iterative algorithm for solving it. Strong convergence of the algorithm to the unique solution of the problem is guaranteed under certain assumptions.

A subgradient-type method for the equilibrium problem over the fixed point set and its applications

In this article, we consider an equilibrium problem: find a point u∈C such that f(u, y) ≥ 0 for all y∈C, where a continuous function satisfies f(x, x) = 0 for all and is a closed convex set. The existing computational methods for this problem require repetitive use of the metric projection onto C, which is often hard to compute. To relax the computational difficulty caused by the metric projection, we present a way to use any firmly nonexpansive mapping T satisfying in place of the metric projection onto C. The proposed method can be applied soundly to the Nash equilibrium problem in noncooperative games.

Fixed point optimization algorithm and its application to network bandwidth allocation

A convex optimization problem for a strictly convex objective function over the fixed point set of a nonexpansive mapping includes a network bandwidth allocation problem, which is one of the central issues in modern communication networks. We devised an iterative algorithm, called a fixed point optimization algorithm, for solving the convex optimization problem and conducted a convergence analysis on the algorithm. The analysis guarantees that the algorithm, with slowly diminishing step-size sequences, weakly converges to a unique solution to the problem. Moreover, we apply the proposed algorithm to a network bandwidth allocation problem and show its effectiveness.

A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping

Many problems to appear in signal processing have been formulated as the variational inequality problem over the fixed point set of a nonexpansive mapping. In particular, convex optimization problems over the fixed point set are discussed, and operators which are considered to the problems satisfy the monotonicity. Hence, the uniqueness of the solution of the problem is not always guaranteed. In this article, we present the variational inequality problem for a monotone, hemicontinuous operator over the fixed point set of a firmly nonexpansive mapping. The main aim of the article is to solve the proposed problem by using an iterative algorithm. To this goal, we present a new iterative algorithm for the proposed problem and its convergence analysis. Numerical examples for the proposed algorithm for convex optimization problems over the fixed point set are provided in the final section.

Acceleration method for convex optimization over the fixed point set of a nonexpansive mapping

The existing algorithms for solving the convex minimization problem over the fixed point set of a nonexpansive mapping on a Hilbert space are based on algorithmic methods, such as the steepest descent method and conjugate gradient methods, for finding a minimizer of the objective function over the whole space, and attach importance to minimizing the objective function as quickly as possible. Meanwhile, it is of practical importance to devise algorithms which converge in the fixed point set quickly because the fixed point set is the set with the constraint conditions that must be satisfied in the problem. This paper proposes an algorithm which not only minimizes the objective function quickly but also converges in the fixed point set much faster than the existing algorithms and proves that the algorithm with diminishing step-size sequences strongly converges to the solution to the convex minimization problem. We also analyze the proposed algorithm with each of the Fletcher–Reeves, Polak–Ribiére–Polyak, Hestenes–Stiefel, and Dai–Yuan formulas used in the conventional conjugate gradient methods, and show that there is an inconvenient possibility that their algorithms may not converge to the solution to the convex minimization problem. We numerically compare the proposed algorithm with the existing algorithms and show its effectiveness and fast convergence.

An Ergodic Algorithm for the Power-Control Games for CDMA Data Networks

In this paper, we consider power control for the uplink of a direct-sequence code-division multiple-access data network. In the uplink, the purpose of power control is for each user to transmit enough power so that it can achieve the required quality of service without causing unnecessary interference to other users in the system. One method that has been very successful in solving this purpose for power control is the game-theoretic approach. The problem for power control is modified as a Nash equilibrium problem in which each user can choose its transmit power in order to maximize its own utility, and a Nash equilibrium is an ideal solution of the power-control game. We present a noncooperative power-control game in which each user can choose the transmit power in a way that it gets the sufficient signal-to-interference-plus-noise ratio and maximizes its own utility. To ensure the existence of a solution, we also propose the variational inequality problem which is connected with the proposed game. On a linear receiver, we deal with the matched filter receiver. Next we present a new ergodic algorithm for the proposed power control because the existing iterative algorithms can not be applied effectively to the proposed power control. We also present convergence analysis for the proposed algorithm. In addition, applying the proposed algorithm to the proposed power control, we provide numerical examples for the transmit power, the signal-to-interference-plus-noise ratio and so on. Numerical results for the proposed algorithm shall show that as compared with the existing power-control game and its method, all users in the network can enjoy the sufficient signal-to-interference-plus-noise ratio and achieve the required quality of service.

Recursive-Rule Extraction Algorithm With J48graft And Applications To Generating Credit Scores

Abstract The purpose of this study was to generate more concise rule extraction from the Recursive-Rule Extraction (Re-RX) algorithm by replacing the C4.5 program currently employed in Re-RX with the J48graft algorithm. Experiments were subsequently conducted to determine rules for six different two-class mixed datasets having discrete and continuous attributes and to compare the resulting accuracy, comprehensibility and conciseness. When working with the CARD1, CARD2, CARD3, German, Bene1 and Bene2 datasets, Re-RX with J48graft provided more concise rules than the original Re-RX algorithm. The use of Re-RX with J48graft resulted in 43.2%, 37% and 21% reductions in rules in the case of the German, Bene1 and Bene2 datasets compared to Re-RX. Furthermore, the Re-RX with J48graft showed 8.87% better accuracy than the Re-RX algorithm for the German dataset. These results confirm that the application of Re-RX in conjunction with J48graft has the capacity to facilitate migration from existing data systems toward new concise analytic systems and Big Data.

Computational Method for Solving a Stochastic Linear-Quadratic Control Problem Given an Unsolvable Stochastic Algebraic Riccati Equation

We discuss a stochastic linear-quadratic control problem in which a stochastic algebraic Riccati equation derived from the problem is unsolvable. The Riccati equation has no solution when the state and control weighting matrices in the objective function of the problem are indefinite, and the conventional methods cannot solve the problem when the Riccati equation itself is unsolvable. We first show that the optimal value of the problem is finite. Next, we formulate a compromise solution to the stochastic algebraic Riccati equation and show that the problem can be solved via this compromise solution under certain assumptions. Moreover, we propose a novel computational method for finding the compromise solution based on iterative techniques for a convex optimization problem over the fixed point set of a certain nonexpansive mapping. Numerical examples demonstrate the effectiveness of this method.