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Convergence rate analysis and error bounds for projection algorithms in convex feasibility problems

OPTIMIZATION METHODS & SOFTWARE 2003年第4期18卷 377-394页

作者： Beck, A Teboulle, M Tel Aviv Univ Sch Math Sci IL-69978 Tel Aviv Israel

We analyze the rate of convergence of three basic projections type algorithms for solving the convex feasibility problem (CFP). Error bounds are known to be central in establishing the rate of convergence of iterative methods. We study the interplay between Slater's hypothesis on CFP and a specific local error bound (LEB). We show that without Slater's hypothesis on CFP, projections type algorithms can in fact behave quite badly, i.e., with a rate of convergence which is not bounded. We derive a new and simple convex analytic proof showing that Slater's hypothesis on CFP implies LEB and hence linear convergence of projection algorithms is guaranteed. We then propose an alternative local error bound derived from the gradient projection algorithm for convex minimization which is proven to be weaker than LEB and used to derive further convergence rate results.

关键词： convex feasibility projection algorithms error bounds convergence rate convex minimization

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Tight l ∞ and l 1 error bounds for projection algorithms in conditional set membership estimation

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IFAC Proceedings Volumes 1999年第2期32卷 4106-4111页

作者： A. Garulli Dipartimento di Ingegneria dell'Informazione Università di Siena Via Roma 56 53100 Siena Italy

In set membership estimation, conditional problems arise when the estimate must belong to a given set of specified structure. Central and interpolatory projection algorithms provide conditional estimates that are suboptimal in terms of the worst-case estimation error. In order to precisely evaluate the suboptimality level of these estimators, tight upper bounds on the estimation error must be computed as a function of the conditional radius of information, which represents the minimum achievable error. In this paper, tight bounds are derived in the V ∞ and V 1 cases, for a general setting which allows to consider any compact set of feasible problem elements and linearly parametrized estimates.

关键词： Set membership estimation identification projection algorithms

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Random projection algorithms for convex set intersection problems

Random projection algorithms for convex set intersection pro...

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IEEE Conference on Decision and Control

作者： A. Nedić Department of Industrial and Enterprise Systems Engineering University of Illinois Urbana IL USA

The focus of this paper is on the set intersection problem for closed convex sets admitting projection operation in a closed form. The objective is to investigate algorithms that would converge (in some sense) if and only if the problem has a solution. To do so, we view the set intersection problem as a stochastic optimization problem of minimizing the “average” residual error of the set collection. We consider a stochastic gradient method as a main tool for investigating the properties of the stochastic optimization problem. We show that the stochastic optimization problem has a solution if and only if the stochastic gradient method is convergent almost surely. We then consider a special case of the method, namely the random projection method, and we analyze its convergence. We show that a solution of the intersection problem exists if and only if the random projection method exhibits certain convergence behavior almost surely. In addition, we provide convergence rate results for the expected residual error.

关键词： Convergence Gradient methods Random variables Markov processes projection algorithms Approximation methods

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Transient and steady-state analysis of filtered-x affine projection algorithms with imperfect secondary path estimate

Transient and steady-state analysis of filtered-x affine pro...

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International Symposium on Image and Signal Processing and Analysis

作者： A. Carini G.L. Sicuranza STI-University of Urbino Italy DEEI-Università di Trieste Italy

This paper provides an analysis based on energy conservation arguments of the transient and steady-state behaviors of two filtered-x affine projection algorithms in presence of an imperfect estimation of the secondary paths of an active noise control system. Very mild assumptions are posed on the system model, which is only required to have a linear dependence of the output from the filter coefficients.

关键词： Transient analysis Steady-state Algorithm design and analysis projection algorithms Finite impulse response filter Active noise reduction Nonlinear filters Convergence Adaptive filters Control systems

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Optimal Constraint Vectors for Set-Membership Proportionate Affine projection algorithms

Optimal Constraint Vectors for Set-Membership Proportionate ...

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IEEE/SP Workshop on Statistical Signal Processing (SSP)

作者： Marcelo J. M. Spelta Wallace A. Martins Federal University of Rio de Janeiro (UFRJ) PEE/COPPE Rio de Janeiro Brazil

Sparsity is an inherent feature of certain practical systems and appears in problems such as channel equalization and echo cancellation. Designed for exploiting the intrinsic structure of sparse environments, while also taking advantage of the data reuse and selection strategies, the set-membership proportionate affine projection algorithm (SM-PAPA) relies on the choice of a constraint vector (CV) that affects the behavior of the adaptive system. Although the selection of this CV has been based on some heuristics, a recent work proposes an optimal CV for the set-membership affine projection algorithm, a particular instance of the SM-PAPA. This paper adopts a convex optimization framework and generalizes the optimal CV concept for the SM-PAPA, allowing its use in sparse systems. Moreover, by using the gradient projection method for solving the related constrained convex problem, this paper demonstrates that the optimal CV can indeed be applied in real-time applications.

关键词： Signal processing algorithms Optimization IP networks Signal processing projection algorithms Convergence Conferences

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A Study on Multi-Constraint Robotic Arm Path Planning and Energy Optimization Based on RRT* and Gradient projection algorithms

A Study on Multi-Constraint Robotic Arm Path Planning and En...

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IEEE International Conference on Artificial Intelligence and Computer Applications (ICAICA)

作者： Yunye Li Jing Luo School of Economics and Management Changsha University of Science & Technology Changsha China

ISBN: (数字)9798350377255

ISBN: (纸本)9798350377262

To optimize the path planning of a six-degree-of-freedom robotic arm in complex environments, this paper proposes a multi-objective optimization method that integrates the gradient projection method and the RRT* algorithm. Based on the D-H parameter method, the kinematic model of the robotic arm is constructed. The gradient projection method is employed to optimize joint trajectories for energy efficiency, while the RRT* algorithm is used to solve the obstacle avoidance problem. On this basis, a multi-objective path planning model is established, aiming to minimize end-effector error and energy consumption. MATLAB simulations verify the effectiveness of the proposed method. Experimental results demonstrate that the method excels in path smoothness, energy optimization, and obstacle avoidance, providing theoretical and technical support for the application of robotic arms in industrial and service fields.

关键词： Energy consumption Service robots Heuristic algorithms Optimization methods End effectors Real-time systems Mathematical models Trajectory Collision avoidance projection algorithms

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New variable step size affine projection algorithms

New variable step size affine projection algorithms

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International Conference on Communications (COMM)

作者： Felix Albu Constantin Paleologu Silviu Ciochina Faculty of Electrical Engineering Valahia University of Targoviste Targoviste Romania Telecommunication Department Politehnica University of Bucharest Bucuresti Romania

In this paper, we propose new variable step size affine projection algorithms whose step sizes are adjusted according to the square of a time-averaging estimate of the autocorrelation of a priori and a posteriori errors. The proposed algorithms have fast convergence, robustness against near-end signal variations (including double-talk) and do not require any a priori information about the acoustic environment. The simulation results indicate the good performance of the proposed algorithms when compared to similar algorithms.

关键词： Speech projection algorithms Adaptive filters Vectors Echo cancellers Convergence

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Detecting Instability Potentials in Regularization for Fast Affine projection algorithms

Detecting Instability Potentials in Regularization for Fast ...

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Asilomar Conference on Signals, Systems & Computers

作者： Heping Ding Institute for Microstructural Sciences National Research Council Canada

In fast affine projection (FAP) adaptation algorithms, it is needed to explicitly or implicitly perform a matrix inversion, during which a small positive regularization factor plays an important role in keeping the algorithm stable and optimized. While existing schemes choose the regularization factor based on certain system criteria not related to the inversion, this paper proposes a simple scheme that dynamically diagnoses the inversion process itself for potentials of instability. This work paves the way for further studies on "minimal regularization and step-size control" technique. A FAP adopting this technique can be compared with FAPs with existing regularization schemes for convergence and steady state performance.

关键词： projection algorithms Financial advantage program Adaptive filters Convergence Stability Noise cancellation Size control Councils Steady-state Adaptive equalizers

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Simplified proportionate affine projection algorithms

Simplified proportionate affine projection algorithms

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International Conference on Systems, Signals and Image Processing, IWSSIP

作者： Felix Albu Faculty of Electrical Engineering Valahia University of Targoviste Romania

In this paper, new efficient proportionate affine projection algorithms are proposed. They use a simplified way of computing the output error vector. It is shown that the simplified approximated memory improved proportionate affine projection algorithm (SAMIPAPA) offers the best compromise between complexity and performance if compared with other proportionate-type APAs. Also, it is shown that, if the logarithmic proportionate scheme is integrated in SAMIPAPA, the convergence speed and tracking abilities are worse.

关键词： Complexity theory Echo cancellers projection algorithms Approximation methods Adaptive filters Vectors Approximation algorithms

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Generalized projection algorithms for identification of mixed uncertainty models

Generalized projection algorithms for identification of mixe...

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IEEE Conference on Decision and Control

作者： L. Giarre A. Garulli G. Zappa Dipt. di Autom. e Inf. Politecnico di Torino Italy DII-Universita degli Studi di Siena Siena Italy USI-Universitáa di Firenze Florence Italy

In this paper the problem of identifying a mixed uncertainty model suitable for robust control, accounting for both unmodelled dynamics and measurement errors, is considered. A generalized projection algorithm estimating the best model consistent with data is determined, solving the problem off-line (via linear programming) and online (via recursive cone algorithm).

关键词： projection algorithms Uncertainty Parametric statistics Robustness Robust control Recursive estimation Nonlinear dynamical systems Measurement errors Delay lines Ear

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