Noriyoshi Sukegawa

Improving bounds on the diameter of a polyhedron in high dimensions

Abstract In 1992, Kalai and Kleitman proved that the diameter of a d -dimensional polyhedron with n facets is at most n 2 + log 2 d . In 2014, Todd improved the Kalai–Kleitman bound to ( n − d ) log 2 d . We improve the Todd bound to ( n − d ) − 1 + log 2 d for n ≥ d ≥ 7 , ( n − d ) − 2 + log 2 d for n ≥ d ≥ 37 , and ( n − d ) − 3 + log 2 d + O 1 ∕ d for n ≥ d ≥ 1 .

Lagrangian relaxation and pegging test for the clique partitioning problem

The clique partitioning problem is an NP-hard combinatorial optimization problem with applications to data analysis such as clustering. Though a binary integer linear programming formulation has been known for years, one needs to deal with a huge number of variables and constraints when solving a large instance. In this paper, we propose a size reduction algorithm which is based on the Lagrangian relaxation and the pegging test, and verify its validity through numerical experiments. We modify the conventional subgradient method in order to manage the high dimensionality of the Lagrangian multipliers, and also make an improvement on the ordinary pegging test by taking advantage of the structural property of the clique partitioning problem.

Fractional programming formulation for the vertex coloring problem

We devise a new formulation for the vertex coloring problem. Different from other formulations, decision variables are associated with pairs of vertices. Consequently, colors will be distinguishable. Although the objective function is fractional, it can be replaced by a piece-wise linear convex function. Numerical experiments show that our formulation has significantly good performance for dense graphs.

A refinement of Todd's bound for the diameter of a polyhedron

Recently, Todd obtained a new bound on the diameter of a polyhedron using an analysis due to Kalai and Kleitman in 1992. In this short note, we prove that the bound by Todd can further be improved. Although our bound is not valid when the dimension is 1 or 2, it is tight when the dimension is 3, and fits better for a high-dimensional polyhedron with a large number of facets.

Cutting plane algorithms for mean-CVaR portfolio optimization with nonconvex transaction costs

This paper studies the mean-risk portfolio optimization problem with nonconvex transaction costs. We employ the conditional value-at-risk (CVaR) as a risk measure. There are a number of studies that aim at efficiently solving large-scale CVaR minimization problems. None of these studies, however, take into account nonconvex transaction costs, which are present in practical situations. To make a piecewise linear approximation of the transaction cost function, we utilized special ordered set type two constraints. Moreover, we devised a subgradient-based cutting plane algorithm to handle a large number of scenarios. This cutting plane algorithm needs to solve a mixed integer linear programming problem in each iteration, and this requires a substantial computation time. Thus, we also devised a two-phase cutting plane algorithm that is even more efficient. Numerical experiments demonstrated that our algorithms can attain near-optimal solutions to large-scale problems in a reasonable amount of time. Especially when rebalancing a current portfolio that is close to an optimal one, our algorithms considerably outperform other solution methods.

Minimum Point-Overlap Labeling

In the air-traffic control, the information related to each air-plane needs to be always displayed as the label. Motivated by this application, de Berg and Gerrits (Comput. Geom. 2012) presented free-label maximization problem, where the goal is to maximize the number of intersection-free labels. In this paper, we introduce an alternative labeling problem for the air-traffic control, called point-overlap minimization. In this problem, we focus on the number of overlapping labels at a point in the plane, and minimize the maximum among such numbers. Instead of maximizing the number of readable labels as in the free-label maximization, we here minimize the cost required for making unreadable labels readable. We provide a 4-approximation algorithm using LP rounding for arbitrary rectangular labels and a faster combinatorial 8-approximation algorithm for unit-square labels.

A Latent-class Model for Estimating Product-choice Probabilities from Clickstream Data

This paper analyzes customer product-choice behavior based on the recency and frequency of each customers page views on e-commerce sites. Recently, we devised an optimization model for estimating product-choice probabilities that satisfy monotonicity, convexity, and concavity constraints with respect to recency and frequency. This shape-restricted model delivered high predictive performance even when there were few training samples. However, typical e-commerce sites deal in many different varieties of product, so the predictive performance of the model can be further improved by integrating such product heterogeneity. For this purpose, we develop a novel latent-class shape-restricted model for estimating product-choice probabilities for each latent class of products. We also give a tailored expectation-maximization algorithm for parameter estimation. Computational results demonstrate that higher predictive performance is achieved with our latent-class model than with the previous shape-restricted model or latent-class logistic regression.

A Simple Projection Algorithm for Linear Programming Problems

Fujishige et al. propose the LP-Newton method, a new algorithm for linear programming problem (LP). They address LPs which have a lower and an upper bound for each variable, and reformulate the problem by introducing a related zonotope. The LP-Newton method repeats projections onto the zonotope by Wolfe’s algorithm. For the LP-Newton method, Fujishige et al. show that the algorithm terminates in a finite number of iterations. Furthermore, they show that if all the inputs are rational numbers, then the number of projections is bounded by a polynomial in L, where L is the input length of the problem. In this paper, we propose a modification to their algorithm using a binary search. In addition to its finiteness, if all the inputs are rational numbers and the optimal value is an integer, then the number of projections is bounded by

Maximizing Barber’s bipartite modularity is also hard

L+1