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FAST gradient algorithm WITH DRY-LIKE FRICTION AND NONMONOTONE LINE SEARCH FOR NONCONVEX OPTIMIZATION PROBLEMS

SIAM JOURNAL ON OPTIMIZATION 2024年第3期34卷 2557-2587页

作者： Nguyen, Lien t. Eberhard, Andrew Yu, Xinghuo Li, Chaojie RMIT Univ Sch Sci Melbourne Vic 3000 Australia RMIT Univ Sch Engn Melbourne Vic 3000 Australia Univ New South Wales Sch Elect Engn & Telecommun Kensington NSW 2033 Australia

In this paper, we propose a fast gradient algorithm for the problem of minimizing a differentiable (possibly nonconvex) function in Hilbert spaces. We first extend the dry friction property for convex functions to what we call the dry-like friction property in a nonconvex setting, and then employ a line search technique to adaptively update parameters at each iteration. Depending on the choice of parameters, the proposed algorithm exhibits subsequential convergence to a critical point or full sequential convergence to an ``approximate"" critical point of the objective function. We also establish the full sequential convergence to a critical point under the Kurdyka--\Lojasiewicz (KL) property of a merit function. Thanks to the parameters' flexibility, our algorithm can reduce to a number of existing inertial gradient algorithms with Hessian damping and dry friction. By exploiting variational properties of the Moreau envelope, the proposed algorithm is adapted to address weakly convex nonsmooth optimization problems. In particular, we extend the result on KL exponent for the Moreau envelope of a convex KL function to a broad class of KL functions that are not necessarily convex nor continuous. Simulation results illustrate the efficiency of our algorithm and demonstrate the potential advantages of combining dry-like friction with extrapolation and line search techniques.

关键词： gradient algorithm dry friction inertial techniques KL property line search linear convergence Moreau envelope nonconvex optimization proximal operator

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A fixed-time gradient algorithm for distributed optimization with inequality constraints

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NEUROCOMPUTING 2023年 532卷 106-113页

作者： He, Xing Wei, Boyu Wang, Hui Southwest Univ Coll Elect & Informat Engn Chongqing Key Lab Nonlinear Circuits & Intelligent Chongqing 400715 Peoples R China Chongqing Normal Univ Sch Math Sci Chongqing 401331 Peoples R China

In this paper, a distributed fixed time gradient algorithm for neurodynamic systems is proposed for solv-ing optimization problems with local inequality constraints. The algorithm is designed using fixed time theory and sliding model control techniques, where each agent has a local objective function known only to itself, and the optimal solution of each local objective function sum can be obtained in a fixed time by the information interaction between neighbors under the condition of local inequality constraints. In addition, the upper bound of the fixed time can be obtained and it is proved theoretically that the upper bound of the fixed time is independent of the initial value. Finally, the stability and effectiveness of the algorithm are verified by numerical examples.(c) 2023 Elsevier B.V. All rights reserved.

关键词： gradient algorithm Fixed time convergence Distributed optimization Load sharing control

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An accelerated proximal gradient algorithm for source-independent waveform inversion

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JOURNAL OF APPLIED GEOPHYSICS 2020年 177卷 104030-104030页

作者： Fu, Hongsun Ma, Mingyue Han, Bo Dalian Maritime Univ Sch Sci Dalian 116026 Peoples R China Harbin Inst Technol Dept Math Harbin 150001 Peoples R China

Full Waveform Inversion (FWI) is a promising method for reconstructing subsurface parameters from the full information content in the seismogram. However, FWI is facing many difficulties in practice. The lack of the source wavelet information, the nonlinearity of the forward problems, and ill-posedness of the formulation all contribute to a pressing need for novel regularization techniques and fast algorithms to speed up and improve inversion results. In this paper, by making use of the variable projection method for nonlinear least squares problems, we develop a source-independent frequency-domain FWI strategy. Specifically, the source wavelet for each frequency is removed automatically using a minimum norm solution between the measured and simulated data. Therefore, the inversion process becomes source independent. In order to overcome the defects of overly smoothed edges caused by the classical Tikhonov regularization, sparsity constrained regularization is applied to FWI based on the ability of curvelets to efficiently represent geophysical images. However, non-differentiability of the objective function make it challenging to find an efficient numerical solution. By using the proximal mapping and the Barzilai-Borwein step size rule, we derive a new accelerated proximal gradient algorithm to handle such non-smooth objective functions. The numerical examples show that very good reconstruction results can be obtained by the proposed algorithm without knowledge of the source. Compared to the gradient descent method with a constant step size, the accelerated algorithm gaves a superior result with much higher resolution. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Full waveform inversion Sparsity constrained regularization Variable projection method Proximal operator gradient algorithm

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Local search gradient algorithm based on functional voxel modeling

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PROGRAMMING AND COMPUTER SOFTWARE 2017年第5期43卷 300-306页

作者： Grigor'ev, S. N. Tolok, A. V. Tolok, N. B. Moscow State Technol Univ Stankin Vadkovskii Per 1 Moscow 127055 Russia Russian Acad Sci VA Trapeznikov Inst Control Sci Ul Profsoyuznaya 65 Moscow 117997 Russia

A principle of construction of search algorithms on local segments of a raster area with the use of graphic images of functional voxel modeling is discussed. An algorithm for constructing gradient lines to organize rules for local search of points is given. A principle of generation of M-images for graphical representation of search control parameters is presented. Examples of the work of the local search gradient algorithm to fill a bounded fragment of a raster area of a specified point are given.

关键词： R-function R-functional modeling voxel local geometrical characteristics the functional voxel method functional voxel model M-image gradient algorithm local function

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Adaptive beamforming of ESPAR antenna based on steepest gradient algorithm

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IEICE TRANSACTIONS ON COMMUNICATIONS 2001年第7期E84B卷 1790-1800页

作者： Cheng, J Kamiya, Y Ohira, T ATR Adapt Commun Res Labs Kyoto 6190288 Japan

Conventional adaptive array antenna processing must access signals on all of the array antenna elements. However, because the low-cost electronically steerable passive array radiator (ESPAR) antenna only has a single-port output, all of the signals on the antenna elements cannot be observed. In this paper, a technique for adaptively controlling the loaded reactances on the passive radiators, thus forming both beam and nulls, is presented for the ESPAR antenna. The adaptive algorithm is based on the steepest gradient theory, where the reactances are sequentially perturbed to determine the gradient vector. Simulations show that the ESPAR antenna can be adaptive. The statistical performance of the output SIR of the ESPAR antenna is also given.

关键词： gradient algorithm adaptive beamforming mobile user terminal interference reduction low complexity

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Matrix rational H₂ approximation:: A gradient algorithm based on schur analysis

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1998年第6期36卷 2103-2127页

作者： Fulcheri, P Olivi, M INRIA F-06902 Sophia Antipolis France

This paper deals with the rational approximation of specified order n to transfer functions which are assumed to be matrix-valued functions in the Hardy space for the complement of the closed unit disk endowed with the L-2-norm. An approach is developed leading to a new algorithm, the first one to our knowledge which concerns matrix-transfer functions in L-2-norm. This approach generalizes the ideas presented in [L. Baratchart, M. Cardelli, and M. Olivi, Automatica, 27(1991), pp. 413-418] in the scalar case but involves substantial new difficulties. Using the Douglas-Shapiro-Shields factorization of transfer functions, the criterion for the rational approximation problem above is expressed in terms of inner matrix functions of McMillan degree n. These functions, which possess a manifold structure, are represented by means of local coordinate maps obtained in [D. Alpay, L. Baratchart, and A. Gombani, Oper. Theory Adv. Appl., 73(1994), pp. 30-66] from a tangential Schur algorithm and for which the coordinates range over n copies of the unit ball. A gradient algorithm is then employed to solve the approximation problem using the coordinate maps to describe the manifold locally and changing from one coordinate map to another when required. However, while processing the gradient algorithm a boundary point can be reached. It is proved that such a point can be considered as an initial point for searching for a local minimum of lower degree while a local minimum of McMillan degree k < n provides a starting point for searching for a local minimum at degree k + 1. The minimization process then pursues through different degrees. The convergence of this algorithm to a local minimum of appropriate degree is proved and demonstrated on a simple example.

关键词： rational approximation identification discrete time systems inner matrices gradient algorithm Schur analysis

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Unbiased plain gradient algorithm for a second-order adaptive IIR notch filter with constrained poles and zeros

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SIGNAL PROCESSING 2010年第8期90卷 2513-2520页

作者： Loetwassana, W. Punchalard, R. Koseeyaporn, J. Wardkein, P. King Mongkuts Inst Technol Ladkrabang Dept Telecommun Engn Bangkok 10520 Thailand Mahanakorn Univ Technol Dept Telecommun Engn Bangkok 10530 Thailand

This article proposes an unbiased plain gradient algorithm for a second-order adaptive IIR notch filter with constrained poles and zeros. The proposed algorithm employs removing a dominant parameter that produces inherent bias. By using this technique, the performances are improved with slight expense in computational complexity. In this paper, theoretical analysis for deriving the estimations of bias and mean square error (MSE) at steady state are presented in closed form. Moreover, the stability bound of the algorithm is also derived. To confirm the analytical results, the computer simulations are provided to corroborate the effectiveness of the proposed algorithm. Furthermore, the performances of the algorithm are also compared with the plain gradient (PG) and modified plain gradient (MPG) algorithms. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Biased and unbiased Adaptive IIR notch filter gradient algorithm

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Steady-state analysis of a plain gradient algorithm for a second-order adaptive IIR notch filter with constrained poles and zeros

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II-ANALOG AND DIGITAL SIGNAL PROCESSING 2001年第7期48卷 733-740页

作者： Xiao, YG Takeshita, Y Shida, K Hiroshima Prefectural Womens Univ Fac Human Life & Environm Sci Minami Ku Hiroshima 7348558 Japan Saga Univ Fac Sci & Engn Saga 8408502 Japan

gradient-type algorithms for adaptive infinite-impulse response (HR) notch filters are very attractive in terms of performance and computation cost as well. However, it is generally quite difficult to assess their performances analytically. There are several trials to analyze adaptive algorithms, such as the sign and the plain gradient algorithms for some types of adaptive HR notch filters, but the analysis techniques used there cannot be directly applied to different types of adaptive HR notch filters. This brief presents closed form expressions for steady-state estimation bias and mean square error (MSE) of a well known plain gradient (LMS-like) second-order adaptive HR notch filter with constrained poles and zeros. First, theoretical expressions for output signals of the notch filter and its corresponding gradient filter at their steady states are developed based on the Taylor series expansions of transfer functions of these two filters in the vicinity of the sinusoidal signal frequency. Difference equations for convergences. in the mean and mean square are then established by using of these output signals, from which the steady-state bias and MSE of the algorithm are derived. Stability bound of the algorithm is also investigated based on the difference equations. Extensive simulations are provided to support the analytical findings.

关键词： adaptive IIR notch filtering constrained notch filter estimation bias estimation mean square error gradient algorithm performance analysis

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A gradient algorithm for the identification of interconnected discrete time varying systems

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ENGINEERING COMPUTATIONS 2020年第3期37卷 909-928页

作者： Ghoul, Yamna Tunis El Manar Univ Natl Engn Sch Tunis Concept & Control Syst Beja Tunisia

Purpose An identification scheme to identify interconnected discrete-time (DT) varying systems. Design/methodology/approach The purpose of this paper is the identification of interconnected discrete time varying systems. The proposed technique permits the division of global system to many subsystems by building a vector observation of each subsystem and then using the gradient method to identify the time-varying parameters of each subsystem. The convergence of the presented algorithm is proven under a given condition. Findings The effectiveness of the proposed technique is then shown with application to a simulation example. Originality/value In the past decade, there has been a renewed interest in interconnected systems that are multidimensional and composed of similar subsystems, which interact with their closest neighbors. In this context, the concept of parametric identification of interconnected systems becomes relevant, as it considers the estimation problem of such systems. Therefore, the identification of interconnected systems is a challenging problem in which it is crucial to exploit the available knowledge about the interconnection structure. For time-varying systems, the identification problem is much more difficult. To cope with this issue, this paper addresses the identification of DT dynamical models, composed by the interconnection of time-varying systems.

关键词： Convergence Discrete-time systems gradient algorithm Interconnected systems Time-varying parameters

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Multi-objective trajectory optimization by a hierarchical gradient algorithm with fuzzy decision logic - Application to slew maneuver problems of a flexible space structure

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TRANSACTIONS OF THE JAPAN SOCIETY FOR AERONAUTICAL AND SPACE SCIENCES 2004年第155期47卷 66-74页

作者： Kojima, H Nakajima, N Tokyo Metropolitan Inst Technol Dept Aerosp Engn Hino Tokyo 191 Japan

The rest-to-rest maneuver problem of a flexible space structure is a two-point boundary value problem (TPBVP) and is solved by some gradient methods. If TPBVP is strongly restricted by constraints, TBVP becomes an ill-defined problem, and the solution meeting all constraints cannot be obtained. However, reasonable suboptimal solutions are often needed since real plant systems are necessary to be controlled. In order to obtain such suboptimal solutions, we have developed a modified version of the hierarchy gradient method by installing fuzzy decision logic. Constraints are classified into non-fuzzy constraints and fuzzy constraints according to their priorities. Fuzzy constraints having a trade-off relationship with each other are compromised reasonably by fuzzy decision logic. The usefulness of the proposed method is numerically and experimentally verified by applying it to the rest-to-rest slew maneuver problem of a flexible space structure, where fuzzy constraints are final time, sensitivity of residual vibration energy with respect to the structure frequency uncertainty and maximum bending moment at the root of the flexible appendage.

关键词： optimal control gradient algorithm fuzzy decision slew maneuver flexible space structure

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