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ON THE APPROXIMATE IDENTIFICATION OF THE LEADING COEFFICIENT OF A LINEAR ELLIPTIC EQUATION BY OUTPUT ERROR AND EQUATION ERROR MINIMIZATION

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OPTIMAL CONTROL APPLICATIONS & METHODS 1989年第3期10卷 229-253页

作者： CROSTA, G Università degli Studi di Milano Dipartimento di Scienze dell' Informazione v. Moretto da B. 9 1-20133 Milan Italy

AbstractBy definition, the approximate solution to the (finite‐dimensional) identification problem is a minimizing element of either cost function. The output error is minimized by two methods: finite difference‐discretized gradient (FD‐DZG) and finite element—discrete gradient (FE‐DG). The equation error is minimized by aDZGmethod. Two‐sided constraints are *** reference to an example of polynomial type in a rectangular domain, some tests are carried out in which the role of several parameters, including data noise, is investigated. Most computational results pertaining to the output error are interpreted according to: (a) sensitivity analysis; (b) an error estimate, which comparesFD‐DZGtoFD‐DG; and (c) a dynamical system model of the *** optimal uniform initial value of the unknown coefficient is found which yields a computed solution very close to the reference solution, regardless of domain discretization and output error minimization method. The superior noise immunity ofFE‐DGoverFD‐DZGis demonstrated. The results obtained from equation error minimization ar

关键词： Coefficient identification Constrained minimization gradient algorithms Inverse problems Linear elliptic equations

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ROBUST PCA METHODS FOR COMPLETE AND MISSING DATA

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NEURAL NETWORK WORLD 2011年第5期21卷 357-392页

作者： Karhunen, Juha Aalto Univ Sch Sci Dept Informat & Comp Sci FI-00076 Espoo Finland

In this paper, we consider and introduce methods for robust principal component analysis (P CA), including also cases where there are missing values in the data. PCA is a widely applied standard statistical method for data preprocessing, compression, and analysis. It is based on the second-order statistics of the data and is optimal for Gaussian data, but it is often applied to data sets having unknown or other types of probability distributions. PCA can be derived from minimization of the mean-square representation error or maximization of variances under orthonormality constraints. However, these quadratic criteria are sensitive to outliers in the data and long-tailed distributions, which may considerably degrade the results given by PCA. We introduce robust methods for estimation of both the PCA eigenvectors directly or the PCA subspace spanned by them. Experimental results show that our methods provide often better results than standard PCA when outliers are present in the data. Furthermore, we extend our methods to incomplete data with missing values. The problems arising in such cases have several features typical for nonlinear models.

关键词： Principal component analysis robustness missing values gradient algorithms imputation method and nearest neighbor method

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Minimization algorithms based on supervisor and searcher cooperation

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2001年第2期111卷 359-379页

作者： Liu, W Dai, YH Univ Kent Canterbury Business Sch Canterbury Kent England Chinese Acad Sci Acad Math & Syst Sci Inst Computat Math & Sci Engn Comp State Key Lab Sci & Engn Comp Beijing Peoples R China

In the present work, we explore a general framework for the design of new minimization algorithms with desirable characteristics, namely, supervisor-searcher cooperation. We propose a class of algorithms within this framework and examine a gradient algorithm in the class. Global convergence is established for the deterministic case in the absence of noise and the convergence rate is studied. Both theoretical analysis and numerical tests show that-the algorithm is efficient for the deterministic case. Furthermore, the fact that there is no line search procedure incorporated in the algorithm seems to strengthen its robustness so that it tackles effectively test problems with stronger stochastic noises. The numerical results for both deterministic and stochastic test problems illustrate the appealing attributes of the algorithm.

关键词： robust algorithms noisy optimization gradient algorithms stochastic approximations

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Distributed Control for Reaching Optimal Steady State in Network Systems: An Optimization Approach

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2018年第3期63卷 864-871页

作者： Zhang, Xuan Papachristodoulou, Antonis Li, Na Harvard Univ Sch Engn & Appl Sci Cambridge MA 02138 USA Harvard Ctr Green Bldg & Cities Cambridge MA 02138 USA Univ Oxford Dept Engn Sci Oxford OX1 3BD England

In this paper, we consider the problem of distributed control for network systems aiming to achieve optimal steady-state performance. Motivated by recent research on reengineering cyber-physical systems, such as power systems and the Internet, we propose a two-step control retrofit procedure. First, we reformulate the dynamical system as an optimization algorithm to solve a certain optimization problem. Second, we combine a predefined steady-state optimization problem and the reformulated problem to systematically (re) design the control. As a result, the system automatically tracks the optimal solution of the predefined steady-state optimization problem and the control scheme can be implemented in a distributed and closed-loop manner. In order to investigate how general this framework is, we establish necessary and sufficient conditions under which a linear dynamical system can be viewed as an optimization algorithm. These conditions are characterized using properties of system matrices and related linear matrix inequalities. A practical example of frequency control in power systems shows the effectiveness of the proposed framework.

关键词： Control retrofit distributed control electric power systems gradient algorithms reverse-engineering

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HEREDITARY OPTIMAL-CONTROL PROBLEMS - NUMERICAL-METHOD BASED UPON A PADE-APPROXIMATION

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1988年第1期56卷 157-166页

作者： LEE, AY STANFORD UNIV PALO ALTOCA 94304

In this paper, we consider a particular approximation scheme which can be used to solve hereditary optimal control problems. These problems are characterized by variables with a time-delayed argumentx(t ? τ). In our approximation scheme, we first replace the variable with an augmented statey(t) ?x(t - τ). The two-sided Laplace transform ofy(t) is a product of the Laplace transform ofx(t) and an exponential factor. This factor is approximated by a first-order Padé approximation, and a differential relation fory(t) can be found. The transformed problem, without any time-delayed argument, can then be solved using a gradient algorithm in the usual way. Four problems are solved to illustrate the validity and usefulness of this technique.

关键词： Padé approximation hereditary optimal control problems nonlinear time-lag systems computational schemes two-sided Laplace transform gradient algorithms

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Distributed robust filtering with H_∞ consensus of estimates

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AUTOMATICA 2011年第1期47卷 1-13页

作者： Ugrinovskii, V. Univ NSW Sch Engn & IT Australian Def Force Acad Canberra ACT 2600 Australia

The paper addresses a problem of design of distributed robust filters using the recent vector dissipativity theory. The main result is a sufficient condition which guarantees a suboptimal H-infinity level of disagreement of estimates in a network of filters. It involves solving a convex optimization/feasibility problem subject to LMI constraints. The special case of balanced interconnection graphs is also considered. A gradient descent type algorithm is presented which allows the nodes to compute their estimator parameters in a decentralized manner. The proposed approach is applied to the problem of observer-based robust synchronization of a nonlinear network to an isolated node. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： Large-scale systems Distributed robust filtering Consensus Vector dissipativity Vector Lyapunov functions gradient algorithms

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Random algorithms for convex minimization problems

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MATHEMATICAL PROGRAMMING 2011年第2期129卷 225-253页

作者： Nedic, Angelia Univ Illinois Dept Ind & Enterprise Syst Engn Urbana IL 61801 USA

This paper deals with iterative gradient and subgradient methods with random feasibility steps for solving constrained convex minimization problems, where the constraint set is specified as the intersection of possibly infinitely many constraint sets. Each constraint set is assumed to be given as a level set of a convex but not necessarily differentiable function. The proposed algorithms are applicable to the situation where the whole constraint set of the problem is not known in advance, but it is rather learned in time through observations. Also, the algorithms are of interest for constrained optimization problems where the constraints are known but the number of constraints is either large or not finite. We analyze the proposed algorithm for the case when the objective function is differentiable with Lipschitz gradients and the case when the objective function is not necessarily differentiable. The behavior of the algorithm is investigated both for diminishing and non-diminishing stepsize values. The almost sure convergence to an optimal solution is established for diminishing stepsize. For non-diminishing stepsize, the error bounds are established for the expected distances of the weighted averages of the iterates from the constraint set, as well as for the expected sub-optimality of the function values along the weighted averages.

关键词： Convex minimization gradient algorithms Subgradient algorithms Random algorithms Error bounds

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FAST ADAPTATION OF EQUALIZER AND DESIRED RESPONSE FOR DIGITAL DATA RECEIVERS

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IEE JOURNAL ON ELECTRONIC CIRCUITS AND SYSTEMS 1978年第5期2卷 159-166页

作者： DOAN, HB CANTONI, A Department of Electrical Engineering University of Newcastle Newcastle Australia

The paper considers the design of the equaliser and the desired overall system impulse response (d.i.r.) for various types of detection algorithms used to recover digital data which has suffered intersymbol interference. The joint optimisation of the equaliser and desired response is formulated using a mean-square error cost. A constraint on the desired response is introduced which enables simple adaptive algorithms to be derived for determining the optimum equalizer and d.i.r. One of the algorithms considered has been found to converge rapidly and is thus ideally suited for application where fast start-up is required. Simulation results of the probability of error performance of various detection algorithms, using the equaliser and d.i.r. designed according to the approach developed in the paper, are presented. These results show that the performance obtained is comparable to that achieved with systems in which the equaliser and d.i.r. are designed according to other techniques. The disadvantage of these other techniques is that they usually lead to algorithms which are difficult to make adaptive and are slow to converge.

关键词： system impulse response General circuit analysis and synthesis methods detection algorithms Data communication equipment and techniques transient response gradient algorithms equalisers digital communication systems signal detection data communication systems simulation studies equaliser design Signal processing and detection intersymbol interference Other data transmission digital data receivers Filters and other networks fast algorithms

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A Donoho-Stark criterion for stable signal recovery in discrete wavelet subspaces

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2011年第17期235卷 5024-5039页

作者： Gosse, Laurent IAC CNR Mauro Picone Sez Bari I-70126 Bari Italy

We derive a sufficient condition by means of which one can recover a scale-limited signal from the knowledge of a truncated version of it in a stable manner following the canvas introduced by Donoho and Stark (1989) [4]. The proof follows from simple computations involving the Zak transform, well-known in solid-state physics. Geometric harmonics (in the terminology of Coifman and Lafon (2006) [22]) for scale-limited subspaces of L-2(R) are also displayed for several test-cases. Finally, some algorithms are studied for the treatment of zero-angle problems. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Product of orthogonal projections Hilbert-Schmidt operator Geometric harmonics Singular operator with closed range gradient algorithms

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An investigation of feasible descent algorithms for estimating the condition number of a matrix

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TOP 2012年第3期20卷 791-809页

作者： Bras, Carmo P. Hager, William W. Judice, Joaquim J. Univ Nova Lisboa CMA Dept Matemat FCT P-2829516 Caparica Portugal Univ Florida Dept Math Gainesville FL 32611 USA Univ Coimbra Dept Matemat P-3001454 Coimbra Portugal Inst Telecomunicacoes P-3001454 Coimbra Portugal

Techniques for estimating the condition number of a nonsingular matrix are developed. It is shown that Hager's 1-norm condition number estimator is equivalent to the conditional gradient algorithm applied to the problem of maximizing the 1-norm of a matrix-vector product over the unit sphere in the 1-norm. By changing the constraint in this optimization problem from the unit sphere to the unit simplex, a new formulation is obtained which is the basis for both conditional gradient and projected gradient algorithms. In the test problems, the spectral projected gradient algorithm yields condition number estimates at least as good as those obtained by the previous approach. Moreover, in some cases, the spectral gradient projection algorithm, with a careful choice of the parameters, yields improved condition number estimates.

关键词： Condition number Numerical linear algebra Nonlinear programming gradient algorithms

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