Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first derivatives are available and that the constraint gradients are sparse.
We discuss an SQP algorithm that uses a smooth augmented Lagrangian merit function and makes explicit provision for infeasibility in the original problem and the QP subproblems. SNOPT is a particular implementation that makes use of a semidefinite QP solver. It is based on a limited-memory quasi-Newton approximation to the Hessian of the Lagrangian and uses a reduced-Hessian algorithm (SQOPT) for solving the QP subproblems. It is designed for problems with many thousands of constraints and variables but a moderate number of degrees of freedom (say, up to 2000). An important application is to trajectory optimization in the aerospace industry. Numerical results are given for most problems in the CUTE and COPS test collections (about 900 examples).

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SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization

Analytic hierarchy process : An overview of applications

In this paper, we will propose a Slacks-Based measure (SBM) of efficiency in DEA. This scalar measure deals directly with the input surplus and the output shortage of the decision making unit (DMU) concerned. It is unit invariant and monotone decreasing with respect to input surplus and output shortage. Furthermore, this measure is decided only by consulting with the reference set of the DMU and is not affected by statistics over the whole data set. The new measure has a close connection with other measures proposed so far, e.g., CCR and BCC. The dual side of this model can be interpreted as profit maximization, in contrast to the ratio maximization of the CCR model. Numerical experiments show its validity as an efficiency measurement tool and its compatibility with other measures of efficiency.

A Slacks Based Measure of Efficiency in Data Envelopment Analysis

A slacks-based measure of super-efficiency in data envelopment analysis

Economists have shown that large and persistent differences in productivity levels across businesses are ubiquitous. This finding has shaped research agendas in a number of fields, including (but not limited to) macroeconomics, industrial organization, labor, and trade. This paper surveys and evaluates recent empirical work addressing the question of why businesses differ in their measured productivity levels. The causes are manifold, and differ depending on the particular setting. They include elements sourced in production practices -- and therefore over which producers have some direct control, at least in theory -- as well as from producers' external operating environments. After evaluating the current state of knowledge, I lay out what I see are the major questions that research in the area should address going forward. (JEL D24, G31, L11, M10, O30, O47)

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What Determines Productivity

In data envelopment analysis (DEA), input and output values are subject to change for several reasons. Such variations differ in their input/output items and their decision-making units (DMUs). Hence, DEA efficiency scores need to be examined by considering these factors. In this paper, we propose new resampling models based on these variations for gauging the confidence intervals of DEA scores. The first model utilizes past-present data for estimating data variations imposing chronological order weights which are supplied by Lucas series (a variant of Fibonacci series). The second model deals with future prospects. This model aims at forecasting the future efficiency score and its confidence interval for each DMU. We applied our models to a dataset composed of Japanese municipal hospitals.

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DEA Scores’ Confidence Intervals with Past-Present and Past-Present-Future Based Resampling

This paper proposes a new search model to deal with public sector decisions in which intangible factors need to be considered along with tangible ones Such problems are often found in a wide range of administrative investigations and criminal inspections in Japan In an effort to manage such problems, this article first constructs an Analytic Hierarchy Process (AHP) model to deal with the intangible factors Then, we reinforce the AHP model by incorporating tangible factors which are not included in the model The probit and/or logit models are applied to test the statistical significance of this combination, based on a past data set This model can be considered as a two-stage procedure in the sense that the AHP results (the first stage) are utilized to construct a statistical model in the second stage, which aims at obtaining a better probability of detection Although this study can foresee the probability of detection for individual objects based on this probability model, there is another important practical issue, ie the scheduling of associated officials Therefore, we propose a scheduling model using mathematical programming methodology Finally, the proposed model was applied to the investigation work of the Post-Clearance Audit Department of Japanese Customs It was found that the probability of finding incorrect declarations would be improved from the current 60% to 75–80% While our problem is not related to criminal activities directly, this study predicts that our approach might be applicable for other governmental investigations and inspections in the scope of this empirical study

A search model with subjective judgments: Auditing of incorrect tax declarations

Xu and Ouenniche (2012a) proposed an input-oriented radial super-efficiency Data Envelopment Analysis (DEA) based model to address a common methodological issue in the evaluation of competing forecasting models; namely, ranking models based on a single performance measure at a time, which typically leads to conflicting ranks. However, their approach suffers from a number of issues. In this paper, we overcome these issues by proposing a slacks-based context-dependent DEA framework and use it to rank forecasting models of oil prices’ volatility.

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Forecasting Models Evaluation using A Slacks-based Context-dependent DEA Framework

In order to express a polyhedron as the Minkowski sum of a polytope and a polyhedral cone, Motzkin (Beitrage zur Theorie der linearen Ungleichungen. Dissertation, University of Basel, 1936) devised a homogenization technique that translates the polyhedron to a polyhedral cone in one higher dimension. Refining his technique, we present a conical representation of a set in the Euclidean space. Then, we use this representation to reach four main results: First, we establish a convex programming based framework for determining a maximal element—an element with the maximum number of positive components—of a non-negative convex set—a convex set in the non-negative Euclidean orthant. Second, we develop a linear programming problem for finding a relative interior point of a polyhedron. Third, we propose two procedures for identifying a strictly complementary solution in linear programming. Finally, we generalize Motzkin’s (Beitrage zur Theorie der linearen Ungleichungen. Dissertation, University of Basel, 1936) representation theorem for a class of closed convex sets in the Euclidean space.

Finding a maximal element of a non-negative convex set through its characteristic cone: an application to finding a strictly complementary solution

Liberalization of the electric power industry has been underway in various countries. Under this situation, it is necessary for electric power companies to improve their management efficiency in order to survive in a competitive market. This study examines management efficiency of vertically integrated electric companies, which consist of several divisions, such as generation, transmission, distribution and so forth. Previous studies mainly focused on efficiency of only a specific division independently. However, we should evaluate firm-level management efficiency based on divisional efficiencies, because these divisions are closely linked each other. For measuring firm-level efficiency, we apply a network DEA model that considers a streamlined relationship among divisions under vertical integration. Furthermore we newly proposed a constrained network DEA model to obtain more practical efficiency index.

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Kaoru Tone

Papers

DEA Scores’ Confidence Intervals with Past-Present and Past-Present-Future Based Resampling

A search model with subjective judgments: Auditing of incorrect tax declarations

Forecasting Models Evaluation using A Slacks-based Context-dependent DEA Framework

Finding a maximal element of a non-negative convex set through its characteristic cone: an application to finding a strictly complementary solution

Application of Network DEA Model to Vertically Integrated Electric Utilities