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Nonlinear model predictive control of functional electrical stimulation

CONTROL ENGINEERING PRACTICE 2017年 58卷 319-331页

作者： Kirsch, Nicholas Alibeji, Naji Sharma, Nitin Univ Pittsburgh Dept Mech Engn & Mat Sci Pittsburgh PA 15261 USA

Minimizing the amount of electrical stimulation can potentially mitigate the adverse effects of muscle fatigue during functional electrical stimulation (FES) induced limb movements. A gradient projection based model predictive controller is presented for optimal control of a knee extension elicited via FES. A control Lyapunov function was used as a terminal cost to ensure stability of the model predictive control. The controller validation results show that the algorithm can be implemented in real-time with a steady-state RMS error of less than 2 degrees. The experiments also show that the controller follows step changes in desired angles and is robust to external disturbances. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Functional electrical stimulation Nonlinear model predictive control Muscle parameter identification Rehabilitation engineering gradient projection algorithm

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Triangular MIMO Relay Channels: Simultaneous Signal and Interference Alignment

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 2015年第1期64卷 223-235页

作者： Teav, Keov Kolyan Zhou, Zhendong Vucetic, Branka Univ Sydney Sch Elect & Informat Engn Sydney NSW 2006 Australia

In this paper, we propose a new three-user network information flow model, referred to as the triangular multiple-input-multiple-output (MIMO) relay (TMR) channel, which consists of three users and three relays equipped with n(U) and n(R) antennas, respectively. Each user sends two independent messages to the other two users via the adjacent relays in two time slots, which are referred to as the multiple-access and broadcast stages. We derive a novel simultaneous signal and interference alignment for the proposed TMR channel in a scenario where there are fewer antennas at each relay than at each user (n(R) < n(U)). An optimized pseudo-inverse scheme based on an efficient gradient projection algorithm is proposed to solve the simultaneous alignment problem. By deriving a gradient over weighted sum-rate maximization and applying a gradient descent method, the optimal beamforming vectors are obtained to maximize the effective signal-to-noise ratios. Furthermore, to obtain rapid convergence speed and reduce computational complexity, we introduce a quasi-Newton method, which is referred to as the Broyden-Fletcher-Goldfarb-Shanno algorithm, by approximating the Hessian matrix of a pure Newton method. The convergence of the proposed gradient algorithm is guaranteed by proposing a line search algorithm. Finally, a performance evaluation shows that the proposed scheme offers a higher sum rate, produces a better outage probability, and achieves a higher multiplexing gain than the existing schemes.

关键词： Beamforming optimization gradient projection algorithm interference alignment multiple-input-multipleoutput (MIMO) relay channel

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Riemannian Newton and trust-region algorithms for analytic rotation in exploratory factor analysis

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BRITISH JOURNAL OF MATHEMATICAL & STATISTICAL PSYCHOLOGY 2021年第1期74卷 139-163页

作者： Liu, Yang Univ Maryland Dept Human Dev & Quantitat Methodol 12308 Benjamin Bldg3942 Campus Dr College Pk MD 20742 USA

In exploratory factor analysis, latent factors and factor loadings are seldom interpretable until analytic rotation is performed. Typically, the rotation problem is solved by numerically searching for an element in the manifold of orthogonal or oblique rotation matrices such that the rotated factor loadings minimize a pre-specified complexity function. The widely used gradient projection (GP) algorithm, although simple to program and able to deal with both orthogonal and oblique rotation, is found to suffer from slow convergence when the number of manifest variables and/or the number of latent factors is large. The present work examines the effectiveness of two Riemannian second-order algorithms, which respectively generalize the well-established truncated Newton and trust-region strategies for unconstrained optimization in Euclidean spaces, in solving the rotation problem. When approaching a local minimum, the second-order algorithms usually converge superlinearly or even quadratically, better than first-order algorithms that only converge linearly. It is further observed in Monte Carlo studies that, compared to the GP algorithm, the Riemannian truncated Newton and trust-region algorithms require not only much fewer iterations but also much less processing time to meet the same convergence criterion, especially in the case of oblique rotation.

关键词： exploratory factor analysis analytic rotation Riemannian optimization matrix manifold truncated Newton algorithm trust-region algorithm conjugate gradient algorithm gradient projection algorithm gradient descent algorithm

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

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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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OPTIMAL TRANSPORTATION UNDER CONTROLLED STOCHASTIC DYNAMICS

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ANNALS OF PROBABILITY 2013年第5期41卷 3201-3240页

作者： Tan, Xiaolu Touzi, Nizar Ecole Polytech CMAP F-91128 Palaiseau France

We consider an extension of the Monge-Kantorovitch optimal transportation problem. The mass is transported along a continuous semimartingale, and the cost of transportation depends on the drift and the diffusion coefficients of the continuous semimartingale. The optimal transportation problem minimizes the cost among all continuous semimartingales with given initial and terminal distributions. Our first main result is an extension of the Kantorovitch duality to this context. We also suggest a finite-difference scheme combined with the gradient projection algorithm to approximate the dual value. We prove the convergence of the scheme, and we derive a rate of convergence. We finally provide an application in the context of financial mathematics, which originally motivated our extension of the Monge-Kantorovitch problem. Namely, we implement our scheme to approximate no-arbitrage bounds on the prices of exotic options given the implied volatility curve of some maturity.

关键词： Mass transportation Kantorovitch duality viscosity solutions 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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Error bound conditions and convergence of optimization methods on smooth and proximally smooth manifolds

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OPTIMIZATION 2022年第3期71卷 711-735页

作者： Balashov, M., V Tremba, A. A. Russian Acad Sci VA Trapeznikov Inst Control Sci Moscow Russia

We analyse the convergence of the gradient projection algorithm, which is finalized with the Newton method, to a stationary point for the problem of nonconvex constrained optimization minx. S f (x) with a proximally smooth set S = {x is an element of R-n : g(x) = 0}, g : R-n -> R-m and a smooth function f. Wepropose new Error bound (EB) conditions for the gradient projection method which lead to the convergence domain of the Newton method. We prove that these EB conditions are typical for a wide class of optimization problems. It is possible to reach high convergence rate of the algorithm by switching to the Newton method.

关键词： Error bound condition gradient projection algorithm Newton's method nonconvex optimization proximal smoothness

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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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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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AN INTERIOR-POINT algorithm FOR LINEARLY CONSTRAINED OPTIMIZATION

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SIAM JOURNAL ON OPTIMIZATION 1992年第3期2卷 450-473页

作者： Wright, Stephen J. Argonne Natl Lab 9700 South Cass Ave Argonne IL 60439 USA

This paper describes an algorithm for optimization of a smooth function subject to general linear constraints. An algorithm of the gradient projection class is used, with the important feature that the "projection" at each iteration is performed by using a primal-dual interior-point method for convex quadratic programming. Convergence properties can be maintained even if the projection is done inexactly in a well-defined way. Higher-order derivative information on the manifold defined by the apparently active constraints can be used to increase the rate of local convergence.

关键词： potential reduction algorithm gradient projection algorithm linearly constrained optimization

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