Full convergence of the steepest descent method with inexact line searches

Burachik, Regina S.; Drummond, L. M. Graña; Iusem, Alfredo N.; Svaiter, B. F.

doi:10.1080/02331939508844042

Cited by 115 publications

(90 citation statements)

References 5 publications

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“…without any additional assumption on boundedness of level sets). Results of this kind for the convex case can be found in [8], [10] and [14] for the method with exogenously given β k 's satisfying (4)-(5), in [12] for the method with exact lineasearches as in (6), and in [2], [7] and [13] for the method with the Armijo rule (7)- (8). We observe that in the case of β k 's given by (4)-(5) the method is not in general a descent one, i.e.…”

Section: The Steepest Descent Methodsmentioning

confidence: 99%

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On the convergence properties of the projected gradient method for convex optimization

Iusem¹

2003

Mat. apl. comput.

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Abstract. When applied to an unconstrained minimization problem with a convex objective, the steepest descent method has stronger convergence properties than in the noncovex case: the whole sequence converges to an optimal solution under the only hypothesis of existence of minimizers (i.e. without assuming e.g. boundedness of the level sets). In this paper we look at the projected gradient method for constrained convex minimization. Convergence of the whole sequence to a minimizer assuming only existence of solutions has also been already established for the variant in which the stepsizes are exogenously given and square summable. In this paper, we prove the result for the more standard (and also more efficient) variant, namely the one in which the stepsizes are determined through an Armijo search. 90C25, 90C30. Mathematical subject classification:

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Section: The Steepest Descent Methodsmentioning

confidence: 99%

“…We prove them in order to make the paper closer to being self-contained. We start with the so called quasi-Fejér convergence theorem (see [7], Theorem 1).…”

Section: Preliminariesmentioning

confidence: 99%

On the convergence properties of the projected gradient method for convex optimization

Iusem¹

2003

Mat. apl. comput.

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“…Note that convergence toward a critical point may appear a weak certificate, but experience in numerical optimization shows that in practice solutions are almost always local minima [8]. The question is now how to construct descent step generators for functions of the form 1 •F and 1,∞ •F .…”

Section: Definitionmentioning

confidence: 99%

“…In synthesis, multipliers and controller variables are updated simultaneously until satisfaction of the FDI in (8). In accordance with [13], we advocate not to use D-K-type methods, where K and are updated alternatingly.…”

Section: Applicationsmentioning

confidence: 99%

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IQC analysis and synthesis via nonsmooth optimization

Apkarian

Noll

2006

Systems & Control Letters

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Integral quadratic constraints (IQCs) are used in system theory to model nonlinear phenomena within the framework of linear feedback control. IQC theory addresses parametric robustness, saturation effects, sector nonlinearities, passivity, and much else. In IQC analysis specially structured linear matrix inequalities (LMIs) arise and are currently addressed by structure exploiting LMI solvers. Controller synthesis under IQC constraints is nonconvex and much harder and has been attempted sporadically by global optimization techniques such as branch and bound, cutting plane or D-K-type coordinate descent ideas. Here, we revisit IQC theory and propose a completely different algorithmic solution based on local and nonsmooth optimization methods. This is less ambitious than global methods, but is very promising in practice. Our approach, while aiming high at IQC synthesis, offers new answers even for IQC analysis, because we optimize without Lyapunov variables. For high-order systems this leads to a significant reduction of the number of unknowns.

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Blind deconvolution by a Newton method on the non‐unitary hypersphere

Fiori

2012

Adaptive Control & Signal

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Blind deconvolution is an inverse filtering technique that has received increasing attention from academia as well as industry because of its theoretical implications and practical applications, such as in speech dereverberation, nondestructive testing and seismic exploration. An effective blind deconvolution technique is known as 'Bussgang', which relies on the iterative Bayesian estimation of the source sequence. Automatic gain control in blind deconvolution keeps constant the energy of the inverse filter impulse response and controls the magnitude of the estimated source sequence. The aim of the present paper is to introduce a class of Newton-type algorithms to optimize the Bussgang cost function on the inverse-filter parameter space whose geometrical structure is induced by the automatic-gain-control constraint. As the parameter space is a differentiable manifold, the Newton-like optimization method is formulated in terms of differential geometrical concepts. The present paper also discusses convergence issues related to the introduced Newton-type optimization algorithms and illustrates their performance on a comparative basis.

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Full convergence of the steepest descent method with inexact line searches

Cited by 115 publications

References 5 publications

On the convergence properties of the projected gradient method for convex optimization

On the convergence properties of the projected gradient method for convex optimization

IQC analysis and synthesis via nonsmooth optimization

Blind deconvolution by a Newton method on the non‐unitary hypersphere

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