Mirai Tanaka

Successive Projection Method for Well-Conditioned Matrix Approximation Problems

Matrices are often required to be well-conditioned in a wide variety of areas including signal processing. Problems to find the nearest positive definite matrix or the nearest correlation matrix that simultaneously satisfy the condition number constraint and sign constraints are presented in this paper. Both problems can be regarded as those to find a projection to the intersection of the closed convex cone corresponding to the condition number constraint and the convex polyhedron corresponding to the other constraints. Thus, we can apply a successive projection method, which is a classical algorithm for finding the projection to the intersection of multiple convex sets, to these problems. The numerical results demonstrated that the algorithm effectively solved the problems.

Efficient preconditioning for noisy separable nonnegative matrix factorization problems by successive projection based low-rank approximations

The successive projection algorithm (SPA) can quickly solve a nonnegative matrix factorization problem under a separability assumption. Even if noise is added to the problem, SPA is robust as long as the perturbations caused by the noise are small. In particular, robustness against noise should be high when handling the problems arising from real applications. The preconditioner proposed by Gillis and Vavasis (SIAM J Optim 25(1):677–698, 2015) makes it possible to enhance the noise robustness of SPA. Meanwhile, an additional computational cost is required. The construction of the preconditioner contains a step to compute the top-k truncated singular value decomposition of an input matrix. It is known that the decomposition provides the best rank-k approximation to the input matrix; in other words, a matrix with the smallest approximation error among all matrices of rank less than k. This step is an obstacle to an efficient implementation of the preconditioned SPA. To address the cost issue, we propose a modification of the algorithm for constructing the preconditioner. Although the original algorithm uses the best rank-k approximation, instead of it, our modification uses an alternative. Ideally, this alternative should have high approximation accuracy and low computational cost. To ensure this, our modification employs a rank-k approximation produced by an SPA based algorithm. We analyze the accuracy of the approximation and evaluate the computational cost of the algorithm. We then present an empirical study revealing the actual performance of the SPA based rank-k approximation algorithm and the modified preconditioned SPA.

A route generation algorithm for an optimal fuel routing problem between two single ports

In this paper, a problem to find the shipping route and speed that minimize the total fuel consumption between two ports is formulated as a mixed-integer nonlinear optimization problem (MINLP). In special cases, the MINLP becomes a mixed-integer second-order cone optimization problem. To solve this problem, the authors propose a route generation algorithm that implicitly enumerates the feasible shipping routes. Enumerating all feasible shipping routes is avoided by computing lower bounds of the optimal value. The effectiveness of our algorithm is verified in numerical experiments.

Box Constrained Low-rank Matrix Approximation with Missing Values.

In this paper, we propose a new low-rank matrix approximation model for completing a matrix with missing values. Our proposed model contains a box constraint that arises from the context of collaborative filtering. Although it is unfortunately NP-hard to solve our model with high accuracy, we can construct a practical algorithm to obtain a stationary point. Our proposed algorithm is based on alternating minimization and converges to a stationary point under a mild assumption.

Mathematical optimization approach for estimating the quantum yield distribution of a photochromic reaction in a polymer

The convolution of a series of events is often observed for a variety of phenomena such as the oscillation of a string. A photochemical reaction of a molecule is characterized by a time constant, but materials in the real world contain several molecules with different time constants. Therefore, the kinetics of photochemical reactions of the materials are usually observed with a complexity comparable with those of theoretical kinetic equations. Analysis of the components of the kinetics is quite important for the development of advanced materials. However, with a limited number of exceptions, deconvolution of the observed kinetics has not yet been mathematically solved. In this study, we propose a mathematical optimization approach for estimating the quantum yield distribution of a photochromic reaction in a polymer. In the proposed approach, time-series data of absorbances are acquired and an estimate of the quantum yield distribution is obtained. To estimate the distribution, we solve a mathematical opti...

Estimation of Passenger Origin-Destination Matrices and Efficiency Evaluation of Public Transportation.

In analyzing and evaluating the efficiency of a transportation system, origin-destination matrices are important. In this paper, we propose an optimization model to estimate origin-destination matrices. The proposed model is easily handled since it is a convex quadratic optimization model. We validate the proposed model using numerical results from real-world data. In addition, we use our model to evaluate the efficiency of a real-world bus service.