Kouichi Taji

A globally convergent Newton method for solving strongly monotone variational inequalities

Variational inequality problems have been used to formulate and study equilibrium problems, which arise in many fields including economics, operations research and regional sciences. For solving variational inequality problems, various iterative methods such as projection methods and the nonlinear Jacobi method have been developed. These methods are convergent to a solution under certain conditions, but their rates of convergence are typically linear. In this paper we propose to modify the Newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length. The purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate. It is then shown that, under the strong monotonicity assumption, the method is globally convergent and, under some additional assumptions, the rate of convergence is quadratic. Limited computational experience indicates the high efficiency of the proposed method.

Unconstrained optimization reformulations of variational inequality problems

Recently, Peng considered a merit function for the variational inequality problem (VIP), which constitutes an unconstrained differentiable optimization reformulation of VIP. In this paper, we generalize the merit function proposed by Peng and study various properties of the generalized function. We call this function the D-gap function. We give conditions under which any stationary point of the D-gap function is a solution of VIP and conditions under which it provides a global error bound for VIP. We also present a descent method for solving VIP based on the D-gap function.

A New Merit Function and A Successive Quadratic Programming Algorithm for Variational Inequality Problems

Recently, various merit functions for variational inequality problems have been proposed and their properties have been studied. Unfortunately, these functions may not be easy to evaluate unless the constraints of the problem have a relatively simple structure. In this paper, a new merit function for variational inequality problems with general convex constraints is proposed. At each point, the proposed function is defined as an optimal value of a quadratic programming problem whose constraints consist of a linear approximation of the given nonlinear constraints. It is shown that the set of constrained minima of the proposed merit function coincides with the set of solutions to the original variational inequality problem. It is also shown that this function is directionally differentiable in all directions and, under suitable assumptions, any stationary point of the function over the constraint set actually solves the original variational inequality problem. Finally, a descent method for solving the variational inequality problem is proposed and its convergence is proved. The method is closely related to a successive quadratic programming method for solving nonlinear programming problems.

Multi-objective capacity planning for agile semiconductor manufacturing

Recent semiconductor business requires an agile fabrication facility (fab) that produces wafers in short cycle time and at low production cost to promote customer satisfaction. In this paper, a multi-objective capacity planning methodology is considered. First, a semiconductor wafer fab is modelled as an open queueing network in which each tool group is a node with a GI/G/m queue. Then, cycle time and production cost are modelled as functions of the number of tools of each tool group and of the throughput of each product. This enables to propose a capacity planning which has a property to quantify the trade-off between cycle time and production cost. The proposed capacity planning can be used to determine a tool-set configuration that satisfies required levels of cycle time and production cost with budget constraint, and to quantify the effect of fab scale on cycle time and production cost. The proposed capacity planning is also applied to the SEMATECH data to show how the following phenomena can be measured. (1) The larger the fab scale, the shorter the cycle time and the lower the production cost. (2) One-machine tool group causes higher production cost or longer waiting time in minifabs. (3) Versatile tools decrease the minimum fab scale that satisfies the required cycle time and production cost.

Biped gait generation based on parametric excitation by knee-joint actuation

Restoring mechanical energy lost by heel-strike collisions is necessary for stable gait generation. One principle to realize this is parametric excitation. Recently, Asano et al. applied this principle to a biped robot with telescopic-legs, and succeeded in generating a sustainable biped gait by computer simulation. In this paper, we deal with a model of a biped robot that has not only semicircular feet but also actuated knees. Though this robot has no actuator at the hip, knee actuators can sustain gait by parametric excitation. We first verify that an actuated knee can cause parametric excitation, and then show by computer simulation that the proposed biped robot can walk continuously with actuated knees only.

Identifying potential repositories for radioactive waste: multiple criteria decision analysis and critical infrastructure systems

An approach for the analysis and management of multiple criteria critical infrastructure problems is put forth. Nuclear waste management involves complex tradeoffs under uncertainty. Among all waste either generated by nature or human activities, radioactive nuclear waste is the most toxic to human health and difficult to manage: it is known that some nuclear waste material will be radioactive and potentially dangerous for hundreds of thousands of years. This paper discusses the use of multiple criteria decision analysis techniques such as the analytic hierarchy process for recommending sites to be considered as potential repositories for nuclear waste.

Parametric Excitation Walking for Four-linked Bipedal Robot

Abstract In the bipedal walking, mechanical energy needs to be restored because of energy lost by heel strike collisions. Harata et al. have applied parametric excitation method to a kneed bipedal robot, and have shown that sustainable gait is generated with only knee torque. In this paper, we propose the method that combines the parametric excitation method for swing-leg with that for support-leg to improve the gait efficiency. To do this, we first extend the three-link bipedal robot to four-link bipedal robot by adding a support-leg knee, and then apply the parametric excitation method to the support-leg. Consequently, parametric excitation method for swing-leg and support-leg restores energy twice a step, and gait of the proposed method grows in efficiency. In the method proposed by Harata et al., the robot has large shin masses to restore much energy and has large semicircular feet to decrease the energy lost by heel strike. These features are unfavorable because common bipedal robots do not have such features. By simulation, we show that the bipedal robot with small shin masses or small feet can walk sustainably because the proposed method increases the quantity of restored energy. For example, the ratio of shin mass to thigh mass is one and the foot radius is reduced to one fifth of the previous method.

Parametric excitation based gait generation for ornithoid walking

The parametric excitation based gait generation method proposed by Asano et al. restores mechanical energy lost by heel-strike collisions. Harata et. al. applied this method to a kneed biped robot which is proper for the parametric excitation, and show that sustainable gait has been generated with only knee torque. A swing-leg of a kneed biped robot has similar mechanism to an acrobot, and many acrobots bends a joint in inverse direction like ornithoid walking. This suggests that inverse bending a knee restores more mechanical energy than forward bending like human walking, and hence, inverse bending may be more efficient. In this paper, we propose a parametric excitation based ornithoid gait generation method for a kneed biped robot, and show that it can walk sustainably by numerical simulation. We also show that parametric excitation based inverse bending walking is more efficient than parametric excitation based forward bending walking with respect to performance indices in our model.

A Globally Convergent Smoothing Newton Method for Nonsmooth Equations and Its Application to Complementarity Problems

The complementarity problem is theoretically and practically useful, and has been used to study and formulate various equilibrium problems arising in economics and engineerings. Recently, for solving complementarity problems, various equivalent equation formulations have been proposed and seem attractive. However, such formulations have the difficulty that the equation arising from complementarity problems is typically nonsmooth. In this paper, we propose a new smoothing Newton method for nonsmooth equations. In our method, we use an approximation function that is smooth when the approximation parameter is positive, and which coincides with original nonsmooth function when the parameter takes zero. Then, we apply Newtons method for the equation that is equivalent to the original nonsmooth equation and that includes an approximation parameter as a variable. The proposed method has the advantage that it has only to deal with a smooth function at any iteration and that it never requires a procedure to decrease an approximation parameter. We show that the sequence generated by the proposed method is globally convergent to a solution, and that, under semismooth assumption, its convergence rate is superlinear. Moreover, we apply the method to nonlinear complementarity problems. Numerical results show that the proposed method is practically efficient.

Parametric excitation-based inverse bending gait generation

In a gait generation method based on the parametric excitation principle, appropriate motion of the center of mass restores kinetic energy lost by heel strike. The motion is realized by bending and stretching a swing-leg regardless of bending direction. In this paper, we first show that inverse bending restores more mechanical energy than forward bending, and then propose a parametric excitation-based inverse bending gait for a kneed biped robot, which improves gait efficiency of parametric excitation walking.