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About the Lipschitz property of the metric projection in the Hilbert space

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2012年第2期394卷 545-551页

作者： Balashov, Maxim V. Golubev, Maxim O. Moscow Inst Phys & Technol Dept Higher Math Dolgoprudnyi 141700 Moscow Region Russia

We consider the metric projection operator from the real Hilbert space onto a strongly convex set. We prove that the restriction of this operator on the complement of some neighborhood of the strongly convex set is Lipschitz continuous with the Lipschitz constant strictly less than 1. This property characterizes the class of strongly convex sets and (to a certain degree) the Hilbert space. We apply the results obtained to the question concerning the rate of convergence for the gradient projection algorithm with differentiable convex function and strongly convex set. (C) 2012 Elsevier Inc. All rights reserved.

关键词： Hilbert space Distance function Metric projection Strongly convex set of radius R Supporting principle Mazur intersection property gradient projection algorithm

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A Schwarz domain decomposition method with gradient projection for optimal control governed by elliptic partial differential equations

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2011年第17期235卷 5078-5094页

作者： Chang, Huibin Yang, Danping E China Normal Univ Dept Math Shanghai 200241 Peoples R China

A domain decomposition method (DDM) is presented to solve the distributed optimal control problem. The optimal control problem essentially couples an elliptic partial differential equation with respect to the state variable and a variational inequality with respect to the constrained control variable. The proposed algorithm, called SA-GP algorithm, consists of two iterative stages. In the inner loops, the Schwarz alternating method (SA) is applied to solve the state and co-state variables, and in the outer loops the gradient projection algorithm (GP) is adopted to obtain the control variable. Convergence of iterations depends on both the outer and the inner loops, which are coupled and affected by each other. In the classical iteration algorithms, a given tolerance would be reached after sufficiently many iteration steps, but more iterations lead to huge computational cost. For solving constrained optimal control problems, most of the computational cost is used to solve PDEs. In this paper, a proposed iterative number independent of the tolerance is used in the inner loops so as to save a lot of computational cost. The convergence rate of L-2-error of control variable is derived. Also the analysis on how to choose the proposed iteration number in the inner loops is given. Some numerical experiments are performed to verify the theoretical results. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Distributed optimal control Elliptic partial differential equation Domain decomposition Schwarz alternating method gradient projection algorithm Geometric convergence rate

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Spectral mixed Galerkin method for state constrained optimal control problem governed by the first bi-harmonic equation

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2011年第14期88卷 2988-3011页

作者： Zhou, Jianwei Yang, Danping E China Normal Univ Dept Math Shanghai 200062 Peoples R China

An optimal control problem governed by the first bi-harmonic equation with the integral constraint for the state and its spectral approximations based on a mixed formulation are investigated. The optimality conditions of the exact and the discrete optimal control systems are derived. The a priori error estimates of high order spectral accuracy are obtained. Furthermore, a simple and efficient iterative algorithm is proposed to solve mixed discrete system. Some numerical examples are performed to verify the theoretical results.

关键词： optimal control state constraint bi-harmonic equation mixed formulation gradient projection algorithm a priori error estimate

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A linearly convergent dual-based gradient projection algorithm for quadratically constrained convex minimization

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MATHEMATICS OF OPERATIONS RESEARCH 2006年第2期31卷 398-417页

作者： Beck, Amir Teboulle, Marc Technion Israel Inst Technol Fac Ind Engn & Management IL-32000 Haifa Israel Tel Aviv Univ Sch Math Sci IL-69978 Tel Aviv Israel

This paper presents a new dual formulation for quadratically constrained convex programs. The special structure of the derived dual problem allows us to apply the gradient projection algorithm to produce a simple expl... 详细信息

关键词： quadratically constrained problems gradient projection algorithm convex minimization rate of convergence analysis

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FINITE ELEMENT APPROXIMATIONS OF AN OPTIMAL CONTROL PROBLEM WITH INTEGRAL STATE CONSTRAINT

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2010年第3期48卷 1163-1185页

作者： Liu, Wenbin Yang, Danping Yuan, Lei Ma, Chaoqun Hunan Univ Sch Business Adm Changsha 410082 Hunan Peoples R China Univ Kent KBS Canterbury CT2 7PE Kent England E China Normal Univ Dept Math Shanghai 200241 Peoples R China SW Forest Univ Dept Fundamental Courses Kunming 650224 Yunnan Peoples R China

An integral state-constrained optimal control problem governed by an elliptic partial differential equation and its finite element approximation are considered. The finite element approximation is constructed on multimeshes. An L-2-norm a priori error estimate of the finite element approximation is obtained. Further, some superconvergence results are proved. Based on these superconvergence results, almost optimal L-infinity-norm error estimates are derived. Some recovery algorithms are then proposed to produce a posteriori error estimators of gradient type. To solve the finite element system, a simple and yet efficient iterative gradient projection algorithm is proposed and its convergence rate is proved. Some numerical examples are performed to confirm theoretical analysis.

关键词： optimal control problem integral state constraint finite element approximation gradient projection algorithm a priori error estimate

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Newton algorithms for analytic rotation: An implicit function approach

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PSYCHOMETRIKA 2008年第2期73卷 231-259页

作者： Boik, Robert J. Montana State Univ Dept Math Sci Bozeman MT 59717 USA

In this paper implicit function-based parameterizations for orthogonal and oblique rotation matrices are proposed. The parameterizations are used to construct Newton algorithms for minimizing differentiable rotation criteria applied to m factors and p variables. The speed of the new algorithms is compared to that of existing algorithms and to that of Newton algorithms based on alternative parameterizations. Several rotation criteria were examined and the algorithms were evaluated over a range of values for m. Initial guesses for Newton algorithms were improved by subconvergence iterations of the gradient projection algorithm. Simulation results suggest that no one algorithm is fastest for minimizing all criteria for all values of m. Among competing algorithms, the gradient projection algorithm alone was faster than the implicit function algorithm for minimizing a quartic criterion over oblique rotation matrices when m is large. In all other conditions, however, the implicit function algorithms were competitive with or faster than the fastest existing algorithms. The new algorithms showed the greatest advantage over other algorithms when minimizing a nonquartic component loss criterion.

关键词： component loss criterion factor analysis gradient projection algorithm oblique rotation orthogonal rotation orthogonal matrix planar algorithm principal components

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Supply chain network equilibrium problem with capacity constraints

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PAPERS IN REGIONAL SCIENCE 2008年第4期87卷 605-622页

作者： Chen, Huey-Kuo Chou, Huey-Wen Natl Cent Univ Dept Civil Engn Wu Chuan Li Jung Li Taiwan Natl Cent Univ Dept Informat Management Wu Chuan Li Jung Li Taiwan

The supply chain network equilibrium problem with capacity constraints (SCNE-C) is an extension of the supply chain network equilibrium problem (SCNE), which also takes into account capacity constraints which refer to the maximum production capacity for a manufacturer or the maximum storage/display space for a retailer. Due to inherent link interactions in the demand functions and cost functions, the SCNE-C problem is formulated as a mathematical model using the variational inequality ( VI) approach. This VI model is characterised by the so-called Wardrop second principle ( in terms of the 'generalised' route cost). To solve the model, a path-based four-loop nested diagonalisation method, along with a supernetwork representation, is proposed and demonstrated with a few numerical examples. The obtained results fully comply with the Wardrop second principle at both retailer sector and demand markets and can provide useful route information of the product. In addition, the stricter the capacity constraints imposed, the lower the quantity demanded will be, and provided at a higher product price. The concepts developed in this paper can be extended into many other spatial price equilibrium problems.

关键词： Supply chain network equilibrium capacity constraint gradient projection algorithm variational inequality approach

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Traffic assignment model considering emissions effect under different engine operating modes

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Wuhan Ligong Daxue Xuebao/Journal of Wuhan University of Technology 2008年第10期30卷 102-104+408页

作者： Xiong, Wei Yan, Xin-Ping Engineering Research Center of Transportation Safety Ministry of Education Wuhan University of Technology Wuhan 430063 China

The traffic assignment problem with nonadditive path costs was discussed, especially considering the emission effect on traditional traffic assignment model. Based on Venigalla's research, the traffic assignment model considering emission at different engine operating modes was completed. A path-based algorithm namely gradient projection (GP) algorithm is used to solve the above mathematic model. The case study shows that the GP algorithm has a rapid convergence speed than FW algorithm. The results also show that GP algorithm can be widely used in solving the traffic assignment problem with nonadditive path costs.

关键词： Emission gradient projection algorithm Nonadditive path costs Traffic assignment

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Noisy independent component analysis as a method of rotating the factor scores

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7th International Conference on Independent Component Analysis and Signal Separation

作者： Unkel, Steffen Trendafilov, Nickolay T. Open Univ Fac Math & Comp Dept Stat Walton Hall Milton Keynes MK7 6AA Bucks England

ISBN: (纸本)9783540744931

Noisy independent component analysis (ICA) is viewed as a method of factor rotation in exploratory factor analysis (EFA). Starting from an initial EFA solution, rather than rotating the loadings towards simplicity, the factors are rotated orthogonally towards independence. An application to Thurstone's box problem in psychometrics is presented using a new data matrix containing measurement error. Results show that the proposed rotational approach to noisy ICA recovers the components used to generate the mixtures quite accurately and also produces simple loadings.

关键词： independent component analysis exploratory factor analysis factor rotation factor scores gradient projection algorithm

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On algorithms for the nonparametric maximum likelihood estimator of the failure function with censored data

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JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 2004年第1期13卷 123-140页

作者： Zhang, Y Jamshidian, M Univ Cent Florida Dept Stat & Actuarial Sci Orlando FL 32816 USA Calif State Univ Fullerton Dept Math Fullerton CA 92834 USA

In this article, we study algorithms for computing the nonparametric maximum likelihood estimator (NPMLE) of the failure function with two types of censored data: doubly censored data and (type 2) interval-censored data. We consider two projection methods, namely the iterative convex minorant algorithm (ICM) and a generalization of the Rosen algorithm (GR) and compare these methods to the well-known EM algorithm. The comparison conducted via simulation studies shows that the hybrid algorithms that alternately use the EM and GR for doubly censored data or, alternately, use the EM and ICM for (type 2) interval-censored data appear to be much more efficient than the EM, especially in large sample situation.

关键词： double censoring EM algorithm gradient projection algorithm interval censoring iterative convex minorant algorithm Rosen method

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