Two derivative-free projection approaches for systems of large-scale nonlinear monotone equations

Ahookhosh, Masoud; Amini, Keyvan; Bahrami, Somayeh

doi:10.1007/s11075-012-9653-z

Cited by 53 publications

(48 citation statements)

References 38 publications

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“…is positive semidefinite, we have that J is also positive semidefinite. [26,34,45,20,1]. Inspired by a pioneer work [34], a class of iterative methods for solving nonlinear (smooth) monotone equations were proposed in recent years [45,20,1].…”

Section: Monotonicity Of Fixed-point Mappingsmentioning

confidence: 99%

A Regularized Semi-Smooth Newton Method with Projection Steps for Composite Convex Programs

Xiao

Wen

et al. 2017

J Sci Comput

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Abstract. The goal of this paper is to study approaches to bridge the gap between first-order and second-order type methods for composite convex programs. Our key observations are: i) Many well-known operator splitting methods, such as forward-backward splitting (FBS) and Douglas-Rachford splitting (DRS), actually define a fixedpoint mapping; ii) The optimal solutions of the composite convex program and the solutions of a system of nonlinear equations derived from the fixed-point mapping are equivalent. Solving this kind of system of nonlinear equations enables us to develop second-order type methods. Although these nonlinear equations may be non-differentiable, they are often semi-smooth and their generalized Jacobian matrix is positive semidefinite due to monotonicity. By combining with a regularization approach and a known hyperplane projection technique, we propose an adaptive semi-smooth Newton method and establish its convergence to global optimality. Preliminary numerical results on ℓ 1 -minimization problems demonstrate that our second-order type algorithms are able to achieve superlinear or quadratic convergence.

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Section: Monotonicity Of Fixed-point Mappingsmentioning

confidence: 99%

A Regularized Semi-Smooth Newton Method with Projection Steps for Composite Convex Programs

Xiao

Wen

et al. 2017

J Sci Comput

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“…There are several possible ways to determine the direction d k satisfying (15). For example, the gradient descent direction and some kind of spectral gradient direction and conjugate gradient directions are satisfying these conditions, see [3,45]. Newton and quasi-Newton directions can satisfy (15) with some more assumptions, see [24,25].…”

Section: New Algorithm and Its Convergencementioning

confidence: 99%

“…This implies that Newton's method can be enhanced to obtain the global convergence, which is the convergence to a stationary point from an arbitrary starting point x 0 that may be far away from it. A globally convergent modification of Newton's method is called damped Newton's method exploiting Newton's direction (3) and a line search similar to that discussed in Algorithm 1. The sequence produced by Algorithm 1 converges to an -solution x * satisfying ∇f (x * ) < , which is by no means sufficient to guarantee that x * is a local minimizer.…”

Section: Introductionmentioning

confidence: 99%

On efficiency of nonmonotone Armijo-type line searches

Ahookhosh

Ghaderi

2017

Applied Mathematical Modelling

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Monotonicity and nonmonotonicity play a key role in studying the global convergence and the efficiency of iterative schemes employed in the field of nonlinear optimization, where globally convergent and computationally efficient schemes are explored. This paper addresses some features of descent schemes and the motivation behind nonmonotone strategies and investigates the efficiency of an Armijotype line search equipped with some popular nonmonotone terms. More specifically, we propose two novel nonmonotone terms, combine them into Armijo's rule and establish the global convergence of sequences generated by these schemes. Furthermore, we report extensive numerical results and comparisons indicating the performance of the nonmonotone Armijo-type line searches using the most popular search directions for solving unconstrained optimization problems. Finally, we exploit the considered nonmonotone schemes to solve an important inverse problem arising in signal and image processing.

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“…Owing to complexity, in the past five decades, numerous algorithms and some software packages in virtue of those powerful algorithms have been developed for solving the nonlinear system of equations. See, for example, [1,[3][4][5][6][7][8][9][10][11][12][13][14][15][16] and the references therein. Nevertheless, in practice, no any algorithm can efficiently solve all the systems of equations arising from sciences and engineering.…”

Section: Introductionmentioning

confidence: 99%

“…Aiming at solution of problem (1), many efficient methods were developed recently. Only by incomplete enumeration, we here mention the trust region method [19], the Newton and the quasi-Newton methods [4][5][6]19], the Gauss-Newton methods [7,8], the Levenberg-Marquardt methods [20][21][22], the derivative-free methods and its modified versions [9][10][11][12][13][14][15][16][23][24][25][26][27], the derivative-free conjugate gradient projection method [14], the modified PRP (Polak-Ribière-Polyak) conjugate gradient method [11], the TPRP method [10], the PRP-type method [28], the projection method [23], the FR-type method [9], and the modified spectral conjugate gradient projection method [13]. Summarily, the spectral Hindawi…”

mentioning

confidence: 99%

A Modified Spectral PRP Conjugate Gradient Projection Method for Solving Large‐Scale Monotone Equations and Its Application in Compressed Sensing

Guo

Wan

2019

Mathematical Problems in Engineering

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In this paper, we develop an algorithm to solve nonlinear system of monotone equations, which is a combination of a modified spectral PRP (Polak-Ribière-Polyak) conjugate gradient method and a projection method. The search direction in this algorithm is proved to be sufficiently descent for any line search rule. A line search strategy in the literature is modified such that a better step length is more easily obtained without the difficulty of choosing an appropriate weight in the original one. Global convergence of the algorithm is proved under mild assumptions. Numerical tests and preliminary application in recovering sparse signals indicate that the developed algorithm outperforms the state-of-the-art similar algorithms available in the literature, especially for solving large-scale problems and singular ones.It has been shown that the solution set of problem (1) is convex if it is nonempty [3]. In addition, throughout the paper, the space is equipped with the Euclidean norm ‖ ⋅ ‖ and the inner product ⟨ , ⟩ = , for , ∈ . Aiming at solution of problem (1), many efficient methods were developed recently. Only by incomplete enumeration, we here mention the trust region method [19], the Newton and the quasi-Newton methods [4][5][6]19], the Gauss-Newton methods [7,8], the Levenberg-Marquardt methods [20][21][22], the derivative-free methods and its modified versions [9][10][11][12][13][14][15][16][23][24][25][26][27], the derivative-free conjugate gradient projection method [14], the modified PRP (Polak-Ribière-Polyak) conjugate gradient method [11], the TPRP method [10], the PRP-type method [28], the projection method [23], the FR-type method [9], and the modified spectral conjugate gradient projection method [13]. Summarily, the spectral Hindawi

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Two derivative-free projection approaches for systems of large-scale nonlinear monotone equations

Cited by 53 publications

References 38 publications

A Regularized Semi-Smooth Newton Method with Projection Steps for Composite Convex Programs

A Regularized Semi-Smooth Newton Method with Projection Steps for Composite Convex Programs

On efficiency of nonmonotone Armijo-type line searches

A Modified Spectral PRP Conjugate Gradient Projection Method for Solving Large‐Scale Monotone Equations and Its Application in Compressed Sensing

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