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Tomonari Kitahara

Publications and source records attributed to Tomonari Kitahara.

10 recordsLinked to original sources
arXiv2026

A Unified Maximum-Margin Framework for Data Envelopment Analysis: Theoretical Foundations and Numerical Evidence

Data Envelopment Analysis (DEA) is a widely used non-parametric method for evaluating the relative efficiency of decision-making units (DMUs). However, the classic Charnes-Cooper-Rhodes (CCR) model often suffers from a lack of discriminatory power, particularly when the number of input and output variables is large relative to the number of DMUs. Under such conditions, many DMUs are evaluated as "CCR-efficient" with a score of unity, making it difficult to rank or distinguish between top performers. To address this limitation, we propose a novel Maximum-Margin DEA (MM-DEA) model that integrates the concept of "margin maximization" -- inspired by structural risk minimization in machine learning -- directly into the efficiency measurement framework. Like the traditional super-efficiency model, the proposed MM-DEA model excludes the target DMU from the reference set used to evaluate it; unlike the super-efficiency model, however, MM-DEA imposes an explicit upper bound on the target DMU's own score, ensuring that every score remains within the standard $[0,1]$ range rather than potentially exceeding one. We theoretically demonstrate that the MM-DEA model is a generalized extension of the CCR model, reducing to the latter when the trade-off parameter $\alpha$ is zero. Furthermore, we show that the fractional programming formulation of MM-DEA can be rigorously transformed into a linear programming (LP) problem, ensuring computational efficiency. In our numerical experiments, MM-DEA produced a unique ranking for a high-dimensional synthetic dataset on which the CCR model failed to discriminate among any of the DMUs. Our results suggest that the "margin" serves as a critical metric for managerial resilience and competitive advantage.

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arXiv2025

Bounds for the number of basic feasible solutions generated by the simplex method with the largest distance rule

In this paper, we analyze the simplex method with the largest distance rule and derive upper bounds on the number of different basic feasible solutions generated. The pivoting rule was proposed by Pan [10], and in some cases, it was reported to be more efficient than the renowned steepest edge rule. We show that the analytical framework developed by Kitahara and Mizuno can be extended to this rule, despite its structural differences from previously studied pivoting rules. The resulting bounds involve a geometric parameter $\beta$ determined by the column norms of the constraint matrix. In addition, our analysis does not require a nondegeneracy assumption.

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arXiv2024

Enhancing Top Efficiency by Minimizing Second-Best Scores: A Novel Perspective on Super Efficiency Models in DEA

In this paper, we reveal a new characterization of the super-efficiency model for Data Envelopment Analysis (DEA). In DEA, the efficiency of each decision making unit (DMU) is measured by the ratio the weighted sum of outputs divided by the weighted sum of inputs. In order to measure efficiency of a DMU, ${\rm DMU}_j$, say, in CCR model, the weights of inputs and outputs are determined so that the effiency of ${\rm DMU}_j$ is maximized under the constraint that the efficiency of each DMU is less than or equal to one. ${\rm DMU}_j$ is called CCR-efficient if its efficiency score is equal to one. It often happens that weights making ${\rm DMU}_j$ CCR-efficient are not unique but form continuous set. This can be problematic because the weights representing CCR-efficiencty of ${\rm DMU}_j$ play an important role in making decisions on its management strategy. In order to resolve this problem, we propose to choose weights which minimize the efficency of the second best DMU enhancing the strength of ${\rm DMU}_j$, and demonstrate that this problem is reduced to a linear programming problem identical to the renowned super-efficiency model. We conduct numerical experiments using data of Japanese commercial banks to demonstrate the advantage of the supper-efficiency model.

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arXiv2023

An Update-and-Stabilize Framework for the Minimum-Norm-Point Problem

We consider the minimum-norm-point (MNP) problem over polyhedra, a well-studied problem that encompasses linear programming. We present a general algorithmic framework that combines two fundamental approaches for this problem: active set methods and first order methods. Our algorithm performs first order update steps, followed by iterations that aim to `stabilize' the current iterate with additional projections, i.e., find a locally optimal solution whilst keeping the current tight inequalities. Such steps have been previously used in active set methods for the nonnegative least squares (NNLS) problem. We bound on the number of iterations polynomially in the dimension and in the associated circuit imbalance measure. In particular, the algorithm is strongly polynomial for network flow instances. Classical NNLS algorithms such as the Lawson-Hanson algorithm are special instantiations of our framework; as a consequence, we obtain convergence bounds for these algorithms. Our preliminary computational experiments show promising practical performance.

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arXiv2018

An oracle-based projection and rescaling algorithm for linear semi-infinite feasibility problems and its application to SDP and SOCP

We point out that Chubanov's oracle-based algorithm for linear programming [5] can be applied almost as it is to linear semi-infinite programming (LSIP). In this note, we describe the details and prove the polynomial complexity of the algorithm based on the real computation model proposed by Blum, Shub and Smale (the BSS model) which is more suitable for floating point computation in modern computers. The adoption of the BBS model makes our description and analysis much simpler than the original one by Chubanov [5]. Then we reformulate semidefinite programming (SDP) and second-order cone programming (SOCP) into LSIP, and apply our algorithm to obtain new complexity results for computing interior feasible solutions of homogeneous SDP and SOCP.

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arXiv2018

A generalization of the steepest-edge rule and its number of simplex iterations for a nondegenerate LP

In this paper, we propose a $p$-norm rule, which is a generalization of the steepest-edge rule, as a pivoting rule for the simplex method. For a nondegenerate linear programming problem, we show upper bounds for the number of iterations of the simplex method with the steepest-edge and $p$-norm rules. One of the upper bounds is given by a function of the number of variables, that of constraints, and the minimum and maximum positive elements in all basic feasible solutions.

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arXiv2017

An extension of Chubanov's algorithm to symmetric cones

In this work we present an extension of Chubanov's algorithm to the case of homogeneous feasibility problems over a symmetric cone K. As in Chubanov's method for linear feasibility problems, the algorithm consists of a basic procedure and a step where the solutions are confined to the intersection of a half-space and K. Following an earlier work by Kitahara and Tsuchiya on second order cone feasibility problems, progress is measured through the volumes of those intersections: when they become sufficiently small, we know it is time to stop. We never have to explicitly compute the volumes, it is only necessary to keep track of the reductions between iterations. We show this is enough to obtain concrete upper bounds to the minimum eigenvalues of a scaled version of the original feasibility problem. Another distinguishing feature of our approach is the usage of a spectral norm that takes into account the way that K is decomposed as simple cones. In several key cases, including semidefinite programming and second order cone programming, these norms make it possible to obtain better complexity bounds for the basic procedure when compared to a recent approach by Pe\~na and Soheili. Finally, in the appendix, we present a translation of the algorithm to the homogeneous feasibility problem in semidefinite programming.

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arXiv2011

Klee-Minty's LP and Upper Bounds for Dantzig's Simplex Method

Kitahara and Mizuno (2010) get two upper bounds for the number of different basic feasible solutions generated by Dantzig's simplex method. The size of the bounds highly depends on the ratio between the maximum and minimum values of all the positive elements of basic feasible solutions. In this paper, we show some relations between the ratio and the number of iterations by using an example of LP, which is a simple variant of Klee-Minty's LP. We see that the ratio for the variant is equal to the number of iterations by Dantzig's simplex method for solving it. This implies that it is impossible to get a better upper bound than the ratio. We also give improved results of the upper bounds.

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arXiv2011

A Bound for the Number of Different Basic Solutions Generated by the Simplex Method

In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems having optimal solutions. The bound is polynomial of the number of constraints, the number of variables, and the ratio between the minimum and the maximum values of all the positive elements of primal basic feasible solutions. When the primal problem is nondegenerate, it becomes a bound for the number of iterations. We show some basic results when it is applied to special linear programming problems. The results include strongly polynomiality of the simplex method for Markov Decision Problem by Ye and utilize its analysis.

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