Inexact Numerical Methods for Inverse Eigenvalue Problems

In this paper, we survey some of the latest development in using inexact Newton-like methods for solving inverse eigenvalue problems. These methods require the solutions of nonsymmetric and large linear systems. One can solve the approximate Jacobian equation by iterative methods. However, iterative methods usually oversolve the prob- lem in the sense that they require far more (inner) iterations than is required for the convergence of the Newton-like (outer) iterations. The inexact methods can reduce or minimize the oversolving problem and improve the e±ciency. The convergence rates of the inexact methods are superlinear and a good tradeo® between the required inner and outer iterations can be obtained

An extension of the Cayley transform method for a parameterized generalized inverse eigenvalue problem

[EN] Since recent studies have shown that the Cayley transform method can be an effective iterative method for solving the inverse eigenvalue problem, in this work, we consider using an extension of it for solving a type of parameterized generalized inverse eigenvalue problem and prove its locally quadratic convergence. This type of inverse eigenvalue problem, which includes multiplicative and additive inverse eigenvalue problems, appears in many applications. Also, we consider the case where the given eigenvalues are multiple. In this case, we describe a modified problem that is not overdetermined and discuss the extension of the Cayley transform method for this modified problem. Finally, to demonstrate the effectiveness of these algorithms, we present some numerical examples to show that the proposed methods are practical and efficient.The authors would like to express their heartfelt thanks to the editor and anonymous referees for their useful comments and constructive suggestions that substantially improved the quality and presentation of this article. This research was developed during a visit of Z.D. to Universitat Politecnica de Valencia. Z.D. would like to thank the hospitality shown by D. Sistemes Informatics i Computacio, Universitat Politecnica de Valencia. J.E.R. was partially supported by the Spanish Agencia Estatal de Investigacion (AEI) under grant TIN2016-75985-P, which includes European Commission ERDF funds. The authors thank Carmen Campos for useful comments on an initial draft of the article.Dalvand, Z.; Hajarian, M.; Román Moltó, JE. (2020). 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An Inexact Cayley Transform Method For Inverse Eigenvalue Problems

The Cayley transform method is a Newton-like method for solving inverse eigenvalue problems. If the problem is large, one can solve the Jacobian equation by iterative methods. However, iterative methods usually oversolve the problem in the sense that they require far more (inner) iterations than is required for the convergence of the Newton (outer) iterations. In this paper, we develop an inexact version of the Cayley transform method. Our method can reduce the oversolving problem and improves the efficiency with respect to the exact version. We show that the convergence rate of our method is superlinear and that a good tradeoff between the required inner and outer iterations can be obtained

An Inexact Successive Quadratic Approximation Method for Convex L-1 Regularized Optimization

We study a Newton-like method for the minimization of an objective function that is the sum of a smooth convex function and an l-1 regularization term. This method, which is sometimes referred to in the literature as a proximal Newton method, computes a step by minimizing a piecewise quadratic model of the objective function. In order to make this approach efficient in practice, it is imperative to perform this inner minimization inexactly. In this paper, we give inexactness conditions that guarantee global convergence and that can be used to control the local rate of convergence of the iteration. Our inexactness conditions are based on a semi-smooth function that represents a (continuous) measure of the optimality conditions of the problem, and that embodies the soft-thresholding iteration. We give careful consideration to the algorithm employed for the inner minimization, and report numerical results on two test sets originating in machine learning