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Nonlinear Regularization Path for Quadratic Loss Support Vector Machines

IEEE TRANSACTIONS ON NEURAL NETWORKS 2011年第10期22卷 1613-1625页

作者： Karasuyama, Masayuki Takeuchi, Ichiro Nagoya Inst Technol Dept Engn Nagoya Aichi 4668555 Japan

Regularization path algorithms have been proposed to deal with model selection problem in several machine learning approaches. These algorithms allow computation of the entire path of solutions for every value of regularization parameter using the fact that their solution paths have piecewise linear form. In this paper, we extend the applicability of regularization path algorithm to a class of learning machines that have quadratic loss and quadratic penalty term. This class contains several important learning machines such as squared hinge loss support vector machine (SVM) and modified Huber loss SVM. We first show that the solution paths of this class of learning machines have piecewise nonlinear form, and piecewise segments between two breakpoints are characterized by a class of rational functions. Then we develop an algorithm that can efficiently follow the piecewise nonlinear path by solving these rational equations. To solve these rational equations, we use rational approximation technique with quadratic convergence rate, and thus, our algorithm can follow the nonlinear path much more precisely than existing approaches such as predictor-corrector type nonlinear-path approximation. We show the algorithm performance on some artificial and real data sets.

关键词： parametric programming rational approximation support vector machines

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Solving generalized fuzzy data envelopment analysis model: a parametric approach

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CENTRAL EUROPEAN JOURNAL OF OPERATIONS RESEARCH 2017年第4期25卷 889-905页

作者： Foroughi, Ali Asghar Shureshjani, Roohollah Abbasi Univ Qom Dept Math Isfahan Old Rd Qom *** Iran

Data envelopment analysis (DEA) is a non-parametric technique to assess the performance of a set of homogeneous decision making units (DMUs) with common crisp inputs and outputs. Regarding the problems that are modelled out of the real world, the data cannot constantly be precise and sometimes they are vague or fluctuating. So in the modelling of such data, one of the best approaches is using the fuzzy numbers. Substituting the fuzzy numbers for the crisp numbers in DEA, the traditional DEA problem transforms into a fuzzy data envelopment analysis (FDEA) problem. Different methods have been suggested to compute the efficiency of DMUs in FDEA models so far but the most of them have limitations such as complexity in calculation, non-contribution of decision maker in decision making process, utilizable for a specific model of FDEA and using specific group of fuzzy numbers. In the present paper, to overcome the mentioned limitations, a new approach is proposed. In this approach, the generalized FDEA problem is transformed into a parametric programming, in which, parameter selection depends on the decision maker's ideas. Two numerical examples are used to illustrate the approach and to compare it with some other approaches.

关键词： Data envelopment analysis Fuzzy numbers GFDEA model parametric programming

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Continuous selection and unique polyhedral representation of solutions to convex parametric quadratic programs

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2007年第2期134卷 177-189页

作者： Spjotvold, J. Tondel, P. Johansen, T. A. Norwegian Univ Sci & Technol Dept Engn Cybernet N-7034 Trondheim Norway SINTEF ICT Trondheim Norway

A method for obtaining continuous solutions to convex quadratic and linear programs with parameters in the linear part of the objective function and right-hand side of the constraints is presented. For parameter values for which the problem has nonunique solutions, the optimizer with the least Euclidean norm is selected. The normal cone optimality condition is utilized to obtain a unique polyhedral representation of the piecewise affine minimizer function.

关键词： parametric programming quadratic programming linear programming continuous solutions

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Piecewise linear solution paths with application to direct weight optimization

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AUTOMATICA 2008年第11期44卷 2745-2753页

作者： Roll, Jacob Linkoping Univ Div Automat Control SE-58183 Linkoping Sweden

Recently, pathfollowing algorithms for parametric optimization problems with piecewise linear solution paths have been developed within the field of regularized regression. This paper presents a generalization of these algorithms to a wider class of problems. It is shown that the approach can be applied to the nonparametric system identification method, Direct Weight Optimization (DWO), and be used to enhance the computational efficiency of this method. The most important design parameter in the DWO method is a parameter (lambda) controlling the bias-variance trade-off, and the use of parametric optimization with piecewise linear solution paths means that the DWO estimates can be efficiently computed for all values of lambda simultaneously. This allows for designing computationally attractive adaptive bandwidth selection algorithms. One such algorithm for DWO is proposed and demonstrated in two examples. 0 2008 Elsevier Ltd. All rights reserved.

关键词： parametric programming Piecewise quadratic programming Function approximation Nonlinear system identification Nonparametric identification

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On the global optimization of sums of linear fractional functions over a convex set

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2004年第1期121卷 19-39页

作者： Benson, HP Univ Florida Warrington Coll Business Adm Gainesville FL 32611 USA

The global optimization of the sum of linear fractional functions has attracted the interest of researchers and practitioners for a number of years. Since these types of optimization problems are nonconvex, various specialized algorithms have been proposed for globally solving these problems. However, these algorithms may be difficult to implement and are usually relatively inaccessible. In this article, we show that, by using suitable transformations, a number of potential and known methods for globally solving these problems become available. These methods are often more accessible and use more standard tools than the customized algorithms proposed to date. They include, for example, parametric convex programming and concave minimization methods.

关键词： global optimization linear fractional functions sum of ratios concave minimization parametric programming

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Optimizing fuzzy p-hub center problem with generalized value-at-risk criterion

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APPLIED MATHEMATICAL MODELLING 2014年第15-16期38卷 3987-4005页

作者： Yang, Kai Liu, Yankui Yang, Guoqing Hebei Univ Coll Math & Comp Sci Risk Management & Financial Engn Lab Baoding 071002 Hebei Peoples R China

In this study, we reduce the uncertainty embedded in secondary possibility distribution of a type-2 fuzzy variable by fuzzy integral, and apply the proposed reduction method to p-hub center problem, which is a nonlinear optimization problem due to the existence of integer decision variables. In order to optimize p-hub center problem, this paper develops a robust optimization method to describe travel times by employing parametric possibility distributions. We first derive the parametric possibility distributions of reduced fuzzy variables. After that, we apply the reduction methods to p-hub center problem and develop a new generalized value-at-risk (VaR) p-hub center problem, in which the travel times are characterized by parametric possibility distributions. Under mild assumptions, we turn the original fuzzy p-hub center problem into its equivalent parametric mixed-integer programming problems. So, we can solve the equivalent parametric mixed-integer programming problems by general-purpose optimization software. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the efficiency of the proposed solution methods. (C) 2014 Elsevier Inc. All rights reserved.

关键词： parametric possibility distribution Uncertainty reduction Hub location problem parametric programming Mixed-integer programming

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Methods of critical value reduction for type-2 fuzzy variables and their applications

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2011年第5期235卷 1454-1481页

作者： Qin, Rui Liu, Yan-Kui Liu, Zhi-Qiang Hebei Univ Coll Math & Comp Sci Baoding 071002 Hebei Peoples R China City Univ Hong Kong Sch Creat Media Hong Kong Hong Kong Peoples R China

A type-2 fuzzy variable is a map from a fuzzy possibility space to the real number space;it is an appropriate tool for describing type-2 fuzziness. This paper first presents three kinds of critical values (CVs) for a regular fuzzy variable (RFV), and proposes three novel methods of reduction for a type-2 fuzzy variable. Secondly, this paper applies the reduction methods to data envelopment analysis (DEA) models with type-2 fuzzy inputs and outputs, and develops a new class of generalized credibility DEA models. According to the properties of generalized credibility, when the inputs and outputs are mutually independent type-2 triangular fuzzy variables, we can turn the proposed fuzzy DEA model into its equivalent parametric programming problem, in which the parameters can be used to characterize the degree of uncertainty about type-2 fuzziness. For any given parameters, the parametric programming model becomes a linear programming one that can be solved using standard optimization solvers. Finally, one numerical example is provided to illustrate the modeling idea and the efficiency of the proposed DEA model. (C) 2010 Elsevier By. All rights reserved.

关键词： Fuzzy possibility theory Type-2 fuzzy variable Critical value Reduction methods Data envelopment analysis parametric programming

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Modelling and explicit model predictive control for PEM fuel cell systems

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CHEMICAL ENGINEERING SCIENCE 2012年第1期67卷 15-25页

作者： Panos, C. Kouramas, K. I. Georgiadis, M. C. Pistikopoulos, E. N. Univ London Imperial Coll Sci Technol & Med Ctr Proc Syst Engn Dept Chem Engn London SW7 2AZ England Univ Western Macedonia Dept Engn Informat & Telecommun Kozani 50100 Greece Parametr Optimizat Solut Ltd London N10 3LR England

We present an analytical dynamic model and a general framework for the optima control design of a PEM fuel cell system. The mathematical model consists of a detailed model for the PEM fuel cell stack and simplified models for the compressor, humidifier and cooling system. The framework features (i) a detailed dynamic process model, (ii) a reduced order approximating model obtained by performing dynamic simulations of the system and (iii) the design of an explicit/multi-parametric model predictive controller. The derived explicit/multi-parametric controller is tested and validated off-line on several operating conditions. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： PEM fuel cell Mathematical modelling Reduced order model Explicit/multi-parametric model predictive control parametric programming General framework

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Embedded explicit model predictive vibration control

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MECHATRONICS 2016年 36卷 54-62页

作者： Takacs, Gergely Batista, Gabriel Gulan, Martin Rohal'-Ilkiv, Boris Slovak Univ Technol Bratislava Fac Mech Engn Inst Automat Measurement & Appl Informat Nam Slobody 17 Bratislava 81231 Slovakia

This article presents an embedded active vibration suppression system featuring real-time explicit model predictive control (EMPC) that is implemented on a microcontroller unit (MCU). The EMPC controller minimizes the tip deflection of an aluminum cantilever beam driven by piezoceramic actuators, gaining its feedback from direct position measurements. The output and input performance of the EMPC method is compared to an analogously tuned positive position feedback (PPF) controller. An extensive analysis is provided on the cycle timing and memory needs of the explicit predictive vibration control scheme. The results demonstrate that the EMPC controller may achieve the same vibration suppression results compared to PPF with less input effort, while inherently respecting process constraints. Furthermore, we show that EMPC task execution timing is comparable in the random access memory (RAM) and read only memory (ROM) alternatives, suggesting that numerous current microcontrollers are suitable for EMPC based active vibration control, in case the prediction model is kept simple. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Vibration control Model predictive control Explicit solution parametric programming Embedded systems Microcontroller unit

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Global stabilization of nonlinear systems with inputs subject to magnitude and rate bounds:: A parametric optimization approach

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2000年第3期39卷 682-706页

作者： Solís-Daun, J Suárez, R Alvarez-Ramírez, J Univ Autonoma Metropolitana Iztapalapa Div Ciencias Basicas & Ingn Mexico City DF Mexico

A bounded feedback control design approach is proposed for the global asymptotic stabilization of a class of nonlinear systems with stable free dynamics. The control inputs and their derivatives are constrained to take values on sets defined by a Cartesian product of eta -dimensional closed balls B-r(eta) (p), which are defined by means of a p-norm and a radius vector parameter r. In order to derive the bounded control stabilizer, the resulting procedure implies that gains (as state-functions) are obtained from the solution to a set of c-parameterized nonlinear programming problems. In general, the resulting closed-loop system could be implicitly defined, i.e., consisting of a system of differential equations plus a set of nonlinear algebraic equations(required to compute the control). Special interest is focused on an important class of homogeneous systems that includes a class of globally asymptotically stabilizable systems by linear feedback and bilinear systems. For those systems, the problem of inputs subject to globally bounded rates is also addressed.

关键词： bounded control bounded control rate nonlinear systems global stabilization parametric programming

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