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Learning Translation Invariant Kernels for Classification

JOURNAL OF MACHINE LEARNING RESEARCH 2010年第4期11卷 1353-1390页

作者： Ghiasi-Shirazi, Kamaledin Safabakhsh, Reza Shamsi, Mostafa Amirkabir Univ Technol Dept Comp Engn Tehran 15914 Iran Amirkabir Univ Technol Fac Math & Comp Sci Tehran 15914 Iran

Appropriate selection of the kernel function, which implicitly defines the feature space of an algorithm, has a crucial role in the success of kernel methods. In this paper, we consider the problem of optimizing a kernel function over the class of translation invariant kernels for the task of binary classification. The learning capacity of this class is invariant with respect to rotation and scaling of the features and it encompasses the set of radial kernels. We show that how translation invariant kernel functions can be embedded in a nested set of sub-classes and consider the kernel learning problem over one of these sub-classes. This allows the choice of an appropriate sub-class based on the problem at hand. We use the criterion proposed by Lanckriet et al. (2004) to obtain a functional formulation for the problem. It will be proven that the optimal kernel is a finite mixture of cosine functions. The kernel learning problem is then formulated as a semi-infinite programming (SIP) problem which is solved by a sequence of quadratically constrained quadratic programming (QCQP) sub-problems. Using the fact that the cosine kernel is of rank two, we propose a formulation of a QCQP sub-problem which does not require the kernel matrices to be loaded into memory, making the method applicable to large-scale problems. We also address the issue of including other classes of kernels, such as individual kernels and isotropic Gaussian kernels, in the learning process. Another interesting feature of the proposed method is that the optimal classifier has an expansion in terms of the number of cosine kernels, instead of support vectors, leading to a remarkable speedup at run-time. As a by-product, we also generalize the kernel trick to complex-valued kernel functions. Our experiments on artificial and real-world benchmark data sets, including the USPS and the MNIST digit recognition data sets, show the usefulness of the proposed method.

关键词： kernel learning translation invariant kernels capacity control support vector machines classification semi-infinite programming

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ASYMPTOTIC APPROXIMATION METHOD AND ITS CONVERGENCE ON semi-infinite programming

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Acta Mathematica Scientia 2006年第1期26卷 17-24页

作者：万仲平王先甲何炬林贾世会 School of Mathematics and Statistics Wuhan University Institute of Systems Engineering Wuhan University

The aim of this article is to discuss an asymptotic approximation model and its convergence for the minimax semi-infinite programming problem. An asymptotic surrogate constraints method for the minimax semi-infinite programming problem is presented by making use of two general discrete approximation methods. Simultaneously, the consistence and the epi-convergence of the asymptotic approximation problem are discussed.

关键词： semi-infinite programming asymptotic approximation convergence

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Global convergence of a robust smoothing SQP method for semi-infinite programming

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2006年第1期129卷 147-164页

作者： Ling, C. Qi, L. Q. Zhou, G. L. Wu, S. Y. Zhejiang Univ Finance & Eco Hangzhou 310012 Peoples R China Natl Cheng Kung Univ Inst Appl Math Tainan 70101 Taiwan

The semi-infinite programming (SIP) problem is a program with infinitely many constraints. It can be reformulated as a nonsmooth nonlinear programming problem with finite constraints by using an integral function. Due to the nondifferentiability of the integral function, gradient-based algorithms cannot be used to solve this nonsmooth nonlinear programming problem. To overcome this difficulty, we present a robust smoothing sequential quadratic programming (SQP) algorithm for solving the nonsmooth nonlinear programming problem. At each iteration of the algorthm, we need to solve only a quadratic program that is always feasible and solvable. The global convergence of the algorithm is established under mild conditions. Numerical results are given.

关键词： smoothing SQP algorithm semi-infinite programming integral functions global convergence

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Efficient algorithm for solving semi-infinite programming problems and their applications to nonuniform filter bank designs

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2006年第11期54卷 4223-4232页

作者： Ho, Charlotte Yuk-Fan Ling, Bingo Wing-Kuen Liu, Yan-Qun Tam, Peter Kwong-Shun Teo, Kok-Lay Queen Mary Univ London Dept Elect Engn London E1 4NS England Kings Coll London Dept Elect Engn Div Engn London WC2R 2LS England RMIT Univ Dept Math & Stat Melbourne Vic 3001 Australia Hong Kong Polytech Univ Dept Elect & Informat Engn Kowloon Hong Kong Peoples R China Curtin Univ Technol Dept Math & Stat Perth WA 6001 Australia

An efficient algorithm for solving semi-infinite programming problems is proposed in this paper. The index set is constructed by adding only one of the most violated points in a refined set of grid points. By applying this algorithm for solving the optimum nonuniform symmetric/antisymmetric linear phase finite-impulse-response (FIR) filter bank design problems, the time required to obtain a globally optimal solution is much reduced compared with that of the previous proposed algorithm.

关键词： dual parameterization nonuniform filter banks semi-infinite programming

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A truncated projected newton-type algorithm for large-scale semi-infinite programming

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SIAM JOURNAL ON OPTIMIZATION 2006年第4期16卷 1137-1154页

作者： Ni, Q Ling, C Qi, LQ Teo, KL Nanjing Univ Aeronaut & Astronaut Dept Math Nanjing 210016 Peoples R China Zhejiang Univ Finance & Econ Sch Informat Hangzhou 310012 Peoples R China Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China Curtin Univ Technol Dept Math & Stat Bentley WA 6102 Australia

In this paper, a truncated projected Newton-type algorithm is presented for solving large-scale semi-infinite programming problems. This is a hybrid method of a truncated projected Newton direction and a modified projected gradient direction. The truncated projected Newton method is used to solve the constrained nonlinear system. In order to guarantee global convergence, a robust loss function is chosen as the merit function, and the projected gradient method inserted is used to decrease the merit function. This algorithm is suitable for handling large-scale problems and possesses superlinear convergence rate. The global convergence of this algorithm is proved and the convergence rate is analyzed. The detailed implementation is discussed, and some numerical tests for solving large-scale semi-infinite programming problems, with examples up to 2000 decision variables, are reported.

关键词： semi-infinite programming Karush-Kuhn-Tucker system large-scale problem

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Generation of Dynamic Motions Under Continuous Constraints: Efficient Computation Using B-Splines and Taylor polynomials

Generation of Dynamic Motions Under Continuous Constraints: ...

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IEEE/RSJ International Conference on Intelligent Robots and Systems

作者： Lengagne, Sebastien Mathieu, Paul Kheddar, Abderrahmane Yoshida, Eiichi CNRS AIST Joint Robot Lab JRL CRT UMI3218 Tsukuba Ibaraki Japan

ISBN: (纸本)9781424466757

This paper proposes a new computation method to solve semi-infinite optimization problems for motion planning of robotic systems. Usually, this problem is solved by means of time-grid discretization of the continuous constraints. Unfortunately, discretization may lead to unsafe motions since there is no guarantee of constraint satisfaction between time samples. First, we show that constraints such as joint position and velocity do not need time-discretization to be checked. Then, we present the computation method based on Taylor polynomials to evaluate more complex constraints over time-intervals. This method also applies to continuous equality constraints, to continuous maximum derivative constraint, and to compute the cost function.

关键词： semi-infinite programming motion optimization Taylor polynomials humanoid robots

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AN UV-ALGORITHM FOR semi-infinite MULTIOBJECTIVE programming

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2006年第1-2期21卷 307-313页

作者： Li-Ping, Pang Jie, Shen Dalian Univ Technol Dept Appl Math Dalian Peoples R China Liaoning Normal Univ Math Coll Dalian Peoples R China

In this paper, we give an application of UV-decomposition method of convex programming to multiobjective programming, and offer a new algorithm for solving semi-infinite multiobjective programming. Finally, the superl... 详细信息

关键词： Multiobjective programming semi-infinite programming superlinear convergence UV-decomposition

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A ROBUST MULTI-CRITERIA FUZZY programming PROBLEM CONSIDERING POSSIBILITY MEASURE

A ROBUST MULTI-CRITERIA FUZZY PROGRAMMING PROBLEM CONSIDERIN...

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1st International Symposium on Intelligent Informatics (ISII 2008)

作者： Hasuike, Takashi Ishii, Hiroaki Osaka Univ Grad Sch Informat Sci & Technol Suita Osaka 5650871 Japan

This paper considers a multi-criteria mathematical decision model including fuzzy variables. As the proposed model is not well-defined due to fuzzy variables, thus to solve them directly and analytically, the possibility measure is introduced. Then, in order to aggregate the multi-criteria objective functions into the single-criterion, the fuzzy compromise programming model is proposed. Furthermore, by introducing the interval of each weight to the objective functions, the proposed model is reformulated as a robust programming problem. Since this problem is a semi-infinite programming problem, the an efficient algorithm. is developed using the linearity of the proposed model. Finally, a numerical example is provided to compare proposed models with the basic model.

关键词： Multi-criteria programming Possibility programming Robust programming semi-infinite programming

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Cutting-set methods for robust convex optimization with pessimizing oracles

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OPTIMIZATION METHODS & SOFTWARE 2009年第3期24卷 381-406页

作者： Mutapcic, Almir Boyd, Stephen Stanford Univ Dept Elect Engn Stanford CA 94305 USA

We consider a general worst-case robust convex optimization problem, with arbitrary dependence on the uncertain parameters, which are assumed to lie in some given set of possible values. We describe a general method for solving such a problem, which alternates between optimization and worst-case analysis. With exact worst-case analysis, the method is shown to converge to a robust optimal point. With approximate worst-case analysis, which is the best we can do in many practical cases, the method seems to work very well in practice, subject to the errors in our worst-case analysis. We give variations on the basic method that can give enhanced convergence, reduce data storage, or improve other algorithm properties. Numerical simulations suggest that the method finds a quite robust solution within a few tens of steps;using warm-start techniques in the optimization steps reduces the overall effort to a modest multiple of solving a nominal problem, ignoring the parameter variation. The method is illustrated with several application examples.

关键词： robust optimization cutting-set methods semi-infinite programming minimax optimization games

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Optimal switching instants for a switched-capacitor DC/DC power converter

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AUTOMATICA 2009年第4期45卷 973-980页

作者： Loxton, R. C. Teo, K. L. Rehbock, V. Ling, W. K. Curtin Univ Technol Dept Math & Stat Perth WA 6845 Australia Kings Coll London Dept Elect Engn London WC2R 2LS England

We consider a switched-capacitor DC/DC power converter with variable switching instants. The determination of optimal switching instants giving low output ripple and strong load regulation is posed as a non-smooth dynamic optimization problem. By introducing a set of auxiliary differential equations and applying a time-scaling transformation, we formulate an equivalent optimization problem with semi-infinite constraints. Existing algorithms can be applied to solve this smooth semi-infinite optimization problem. The existence of an optimal solution is also established. For illustration, the optimal switching instants for a practical switched-capacitor DC/DC power converter are determined using this approach. (C) 2008 Elsevier Ltd. All rights reserved.

关键词： Power converter Impulsive dynamical system Switched linear system semi-infinite programming Control parametrization enhancing transform

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