Hiroshige Dan

Convergence Properties of the Inexact Levenberg-Marquardt Method under Local Error Bound Conditions

In this paper, we consider convergence properties of the Levenberg-Marquardt method for solving nonlinear equations. It is well-known that the nonsingularity of Jacobian at a solution guarantees that the Levenberg-Marquardt method has a quadratic rate of convergence. Recently, Yamashita and Fukushima showed that the Levenberg-Marquardt method has a quadratic rate of convergence under the local error bound assumption, which is milder than the nonsingularity of Jacobian. In this paper, we show that the inexact Levenberg-Marquardt method (ILMM), which does not require computing exact search directions, has a superlinear rate of convergence under the same local error bound assumption. Moreover, we propose the ILMM with Armijos stepsize rule that has global convergence under mild conditions.

On the identification of degenerate indices in the nonlinear complementarity problem with the proximal point algorithm

Abstract.In this paper we focus on the problem of identifying the index sets P(x):={i|xi>0}, N(x):={i|Fi(x)>0} and C(x):={i|xi=Fi(x)=0} for a solution x of the monotone nonlinear complementarity problem NCP(F). The correct identification of these sets is important from both theoretical and practical points of view. Such an identification enables us to remove complementarity conditions from the NCP and locally reduce the NCP to a system which can be dealt with more easily. We present a new technique that utilizes a sequence generated by the proximal point algorithm (PPA). Using the superlinear convergence property of PPA, we show that the proposed technique can identify the correct index sets without assuming the nondegeneracy and the local uniqueness of the solution.

A Superlinearly Convergent Algorithm for the Monotone Nonlinear Complementarity Problem Without Uniqueness and Nondegeneracy Conditions

The purpose of this paper is to present an algorithm for solving the monotone nonlinear complementarity problem (NCP) that enjoys superlinear convergence in a genuine sense without the uniqueness and nondegeneracy conditions. Recently, Yamashita and Fukushima (2001) proposed a method based on the proximal point algorithm (PPA) for monotone NCP. The method has the favorable property that a generated sequence converges to the solution set of NCP superlinearly. However, when a generated sequence converges to a degenerate solution, the subproblems may become computationally expensive and the method does not have genuine superlinear convergence. More recently, Yamashita et al. (2001) presented a technique to identify whether a solution is degenerate or not. Using this technique, we construct a differentiable system of nonlinear equations in which the solution is a solution of the original NCP. Moreover, we propose a hybrid algorithm that is based on the PPA and uses this system. We show that the proposed algorithm has a genuine quadratic or superlinear rate of convergence even if it converges to a solution that is neither unique nor nondegenerate.

Optimal Scheduling of Tunnel Inspection and Monitoring for Leveling Yearly Budget

Many infrastructures were built in 1950’s and 60’s intensively in Japan. They got older in half a century, and now it is an urgent task to inspect, monitor and maintain such aging infrastructures. The Ministry of Land, Infrastructure, Transport and Tourism (MLIT) renewed the guideline of tunnel inspection and monitoring in 2014. In this research, we proposed optimization models to level the yearly budget for tunnel inspections and monitoring activities under the MLIT’s guideline. It is very important to resolve the bias of the cost of inspection and monitoring from year to year because Japanese local governments adopt the single-year budget system and would like to set aside the constant budget for inspection and monitoring each year. We conducted some numerical experiments with the practical data of tunnels and confirmed the effectiveness of our proposed models.

Volume computation of the monument of Queen Khentkawess

Giza Plateau Mapping Project (GPMP) directed by Egyptologist Dr. Mark Lehnar achieved digital modeling of the monument of Queen Khentkawess, as known as the fourth pyramid in Giza. As members of GPMP, the authors report how this digital modeling and volume computation of it were done. First, the monument of Queen Khentkawess was scanned by laser radars with 5[mm] mesh accuracy in average. Some parts of wall of the monument on which hieroglyphs appear were scanned separately with a very high-accuracy (0.1[mm] mesh) laser scanner. Second, these scanned raw data were combined, aligned, and integrated into complete 3-D geometry model on a computer. Finally, computation of volumes of the monument was done from the geometry data of the monument, so that archaeologists could use for investigating relationship among the monument and Great Pyramids, Sphinx, and other temples around Giza Plateau. For this purpose the authors have developed a novel approach for computing volume from laser-scanned set of points (point cloud).