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The Bound-Constrained Conjugate Gradient Method for Non-negative Matrices

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2014年第3期162卷 931-953页

作者： Vollebregt, Edwin A. H. Delft Univ Technol DIAM NL-2600 GA Delft Netherlands VORtech BV NL-2600 AG Delft Netherlands

Existing conjugate gradient (CG)-based methods for convex quadratic programs with bound constraints require many iterations for solving elastic contact problems. These algorithms are too cautious in expanding the active set and are hampered by frequent restarting of the CG iteration. We propose a new algorithm called the Bound-Constrained Conjugate Gradient method (BCCG). It combines the CG method with an active-set strategy, which truncates variables crossing their bounds and continues (using the Polak-RibiSre formula) instead of restarting CG. We provide a case with n=3 that demonstrates that this method may fail on general cases, but we conjecture that it always works if the system matrix A is non-negative. Numerical results demonstrate the effectiveness of the method for large-scale elastic contact problems.

关键词： Quadratic program Bound constraints Conjugate gradients method active set algorithm Elastic contact problem

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PMSM Drive system Control Based on Improved active set algorithm

PMSM Drive system Control Based on Improved Active Set Algor...

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20th Chinese Control and Decision Conference

作者： Chen, Shujin Wu, Mingfang Lai, Zhongmin Jiangsu Univ Sci & Technol Prov Key Lab Adv Welding Technol Zhenjiang 212003 Peoples R China

ISBN: (纸本)9781424417339

In order to obtain excellent varied speed performance of welding robot, the predictive model of motor drive system was established. On the basis of the analysis of the permanent magnet synchronous motor drive system, generalized predictive control method was adopted to control the rotor speed and position of the rotor. Because of the input and output constraint of the system, convex quadratic problem was proposed, and then an active set algorithm was adopted to deal with these constrained input and output. At last, the performance of this method is demonstrated by a real-time implementation using a digital signal processor (DSP) chip on a permanent-magnet synchronous motor with sinusoidal flux distribution. Theoretic analysis and experiments verify the validity and feasibility of this approach.

关键词： Generalized Predictive Control active set algorithm Motor Drive System Welding Robot

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Structured Variable Selection with Sparsity-Inducing Norms

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JOURNAL OF MACHINE LEARNING RESEARCH 2011年第10期12卷 2777-2824页

作者： Jenatton, Rodolphe Audibert, Jean-Yves Bach, Francis Ecole Normale Super INRIA SIERRA Project Team Lab Informat INRIA ENS CNRS UMR 8548 F-75214 Paris France Univ Paris Est Imagine ENPC CSTB Ecole Normale Super Lab InformatINRIA ENS CNRS UMR 8548 F-77455 Marne La Vallee France

We consider the empirical risk minimization problem for linear supervised learning, with regularization by structured sparsity-inducing norms. These are defined as sums of Euclidean norms on certain subsets of variables, extending the usual l(1)-norm and the group l(1)-norm by allowing the subsets to overlap. This leads to a specific set of allowed nonzero patterns for the solutions of such problems. We first explore the relationship between the groups defining the norm and the resulting nonzero patterns, providing both forward and backward algorithms to go back and forth from groups to patterns. This allows the design of norms adapted to specific prior knowledge expressed in terms of nonzero patterns. We also present an efficient active set algorithm, and analyze the consistency of variable selection for least-squares linear regression in low and high-dimensional settings.

关键词： sparsity consistency variable selection convex optimization active set algorithm

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Nonparametric Regression on a Graph

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JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 2011年第2期20卷 432-447页

作者： Kovac, Arne Smith, Andrew D. A. C. Univ Bristol Dept Math Bristol BS8 1TW Avon England

The 'Signal plus Noise' model for nonparametric regression can be extended to the case of observations taken at the vertices of a graph. This model includes many familiar regression problems. This article discusses the use of the edges of a graph to measure roughness in penalized regression. Distance between estimate and observation is measured at every vertex in the L-2 norm, and roughness is penalized on every edge in the L-1 norm. Thus the ideas of total variation penalization can be extended to a graph. The resulting minimization problem presents special computational challenges, so we describe a new and fast algorithm and demonstrate its use with examples. The examples include image analysis, a simulation applicable to discrete spatial variation, and classification. In our examples, penalized regression improves upon kernel smoothing in terms of identifying local extreme values on planar graphs. In all examples we use fully automatic procedures for setting the smoothing parameters. Supplemental materials are available online.

关键词： active set algorithm Image analysis Penalized regression Total variation

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A new active set algorithm for box constrained optimization

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SIAM JOURNAL ON OPTIMIZATION 2006年第2期17卷 526-557页

作者： Hager, William W. Zhang, Hongchao Univ Florida Dept Math Gainesville FL 32611 USA

An active set algorithm (ASA) for box constrained optimization is developed. The algorithm consists of a nonmonotone gradient projection step, an unconstrained optimization step, and a set of rules for branching between the two steps. Global convergence to a stationary point is established. For a nondegenerate stationary point, the algorithm eventually reduces to unconstrained optimization without restarts. Similarly, for a degenerate stationary point, where the strong second-order sufficient optimality condition holds, the algorithm eventually reduces to unconstrained optimization without restarts. A specific implementation of the ASA is given which exploits the recently developed cyclic Barzilai - Borwein (CBB) algorithm for the gradient projection step and the recently developed conjugate gradient algorithm CG_DESCENT for unconstrained optimization. Numerical experiments are presented using box constrained problems in the CUTEr and MINPACK-2 test problem libraries.

关键词： nonmonotone gradient projection box constrained optimization active set algorithm ASA cyclic BB method CBB conjugate gradient method CG_DESCENT degenerate optimization

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Least squares and shrinkage estimation under bimonotonicity constraints

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STATISTICS AND COMPUTING 2010年第2期20卷 177-189页

作者： Beran, Rudolf Duembgen, Lutz Univ Bern Bern Switzerland Univ Calif Davis Davis CA 95616 USA

In this paper we describe active set type algorithms for minimization of a smooth function under general order constraints, an important case being functions on the set of bimonotone rxs matrices. These algorithms can be used, for instance, to estimate a bimonotone regression function via least squares or (a smooth approximation of) least absolute deviations. Another application is shrinkage estimation in image denoising or, more generally, regression problems with two ordinal factors after representing the data in a suitable basis which is indexed by pairs (i,j)a{1,aEuro broken vertical bar,r}x{1,aEuro broken vertical bar,s}. Various numerical examples illustrate our methods.

关键词： active set algorithm Dynamic programming Estimated risk Pool-adjacent-violators algorithm Regularization

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Inference and Modeling with Log-concave Distributions

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STATISTICAL SCIENCE 2009年第3期24卷 319-327页

作者： Walther, Guenther Stanford Univ Dept Stat Stanford CA 94305 USA

Log-concave distributions are an attractive choice for modeling and inference, for several reasons: The class of log-concave distributions contains most of the commonly used parametric distributions and thus is a rich and flexible nonparametric class of distributions. Further, the MLE exists and can be computed with readily available algorithms. Thus, no tuning parameter, such as a bandwidth, is necessary for estimation. Due to these attractive properties, there has been considerable recent research activity concerning the theory and applications of log-concave distributions. This article gives a review of these results.

关键词： Nonparametric density estimation shape constraint log-concave density Polya frequency function strongly unimodal iterative convex minorant algorithm active set algorithm

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Quadratic programming with transaction costs

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COMPUTERS & OPERATIONS RESEARCH 2008年第1期35卷 18-33页

作者： Best, Michael J. Hlouskova, Jaroslava Univ Waterloo Fac Math Dept Combinator & Optimizat Waterloo ON N2L 3G1 Canada Inst Adv Studies Dept Econ & Finance Vienna Austria

We consider the problem of maximizing the mean-variance utility function of n assets. Associated with a change in an asset's holdings from its current or target value is a transaction cost. These must be accounted for in practical problems. A straightforward way of doing so results in a 3n-dimensional optimization problem with 3n additional constraints. This higher dimensional problem is computationally expensive to solve. We present an algorithm for solving the 3n-dimensional problem by modifying an active set quadratic programming (QP) algorithm to solve the 3n-dimensional problem as an n-dimensional problem accounting for the transaction costs implicitly rather than explicitly. The method is based on deriving the optimality conditions for the higher dimensional problem solely in terms of lower dimensional quantities and requires substantially less computational effort than any active set QP algorithm applied directly on a 3n-dimensional problem. (C) 2006 Elsevier Ltd. All rights reserved.

关键词： quadratic programming active set algorithm portfolio optimization transaction costs

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The primal-dual active set method for a crack problem with non-penetration

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IMA JOURNAL OF APPLIED MATHEMATICS 2004年第1期69卷 1-26页

作者： Hintermüller, M Kovtunenko, VA Kunisch, K Graz Univ Inst Math A-8010 Graz Austria MA Lavrentyev Hydrodynam Inst Novosibirsk 630090 Russia

The Lame problem in a 2D domain with a crack under a non-penetration condition is considered as a variational inequality. A primal-dual active set method is proposed as an efficient numerical solution technique and compared to a previously employed iterative method for a penalized formulation. Sufficient conditions for monotonic convergence of a discretized version of the proposed algorithm are given and numerical experiments are presented.

关键词： crack non-penetration variational inequality active set algorithm numerical simulation

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An approximate and computationally efficient algorithm for computing reference stress for creep life assessment

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INTERNATIONAL JOURNAL OF PRESSURE VESSELS AND PIPING 1998年第6期75卷 459-465页

作者： Zarrabi, K Modarres-Motlagh, A Univ New S Wales Sch Mech & Mfg Engn Sydney NSW 2052 Australia

One of the powerful methods that are currently available for the design and life assessment of components that operate within the creep range is the reference stress (RS) method. However, for problems for which the RS is not available from existing solutions, one usually needs to use a non-linear finite element method which is normally iterative, time-consuming and computationally expensive. An efficient and effective method for computing an approximate value for RS is described that combines a lower-bound theorem and finite element discretization. The resulted quadratic programming is solved by an active set algorithm. The verification and application of the proposed method are also described. (C) 1998 Elsevier Science Ltd. All rights reserved.

关键词： creep-life assessment reference stress active set algorithm

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