A Globally Convergent Inexact Newton Method for Systems of Monotone Equations

Solodov, M. V.; Svaiter, B. F.

doi:10.1007/978-1-4757-6388-1_18

Cited by 223 publications

(234 citation statements)

References 12 publications

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“…Another extension using certain enlargements (outer approximations) of the operator defining the problem can be found in [28]. Using also the linesearch technique of [26], the framework of [31,28] led to the development of truly globally convergent inexact Newton methods for monotone equations [30] and complementarity problems [25]. However, by itself, the method of [31] does not attain the goal of the present paper.…”

Section: Inexact Proximal Point Iterationsmentioning

confidence: 99%

Forcing strong convergence of proximal point iterations in a Hilbert space

Solodov¹,

Svaiter²

2000

Math. Program.

292

151

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Abstract. This paper concerns with convergence properties of the classical proximal point algorithm for finding zeroes of maximal monotone operators in an infinite-dimensional Hilbert space. It is well known that the proximal point algorithm converges weakly to a solution under very mild assumptions. However, it was shown by Güler [11] that the iterates may fail to converge strongly in the infinite-dimensional case. We propose a new proximal-type algorithm which does converge strongly, provided the problem has a solution. Moreover, our algorithm solves proximal point subproblems inexactly, with a constructive stopping criterion introduced in [31]. Strong convergence is forced by combining proximal point iterations with simple projection steps onto intersection of two halfspaces containing the solution set. Additional cost of this extra projection step is essentially negligible since it amounts, at most, to solving a linear system of two equations in two unknowns.

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Section: Inexact Proximal Point Iterationsmentioning

confidence: 99%

Forcing strong convergence of proximal point iterations in a Hilbert space

Solodov¹,

Svaiter²

2000

Math. Program.

292

151

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“…In other words, (19) is weaker than (20) or (21). In 1998, the literature [12] showed that their proposed method converged superlinearly when the underlying function F is differentiable with ∇F (x * ) nonsingular and ∇F is locally Lipschitz continuous. It is known that the local error bound condition given in (18) (19) hold, then there is a constant ω > 0 such that for sufficiently large k,…”

Section: Convergence Ratementioning

confidence: 99%

“…In the proof of global convergence of our method, the underlying mapping F need only to satisfy the property (2) which is much weaker than monotone or more generally pseudomontone. Moreover, under weaker assumptions than those of [12,15,14], the local rate of convergence of the iterative sequence is established.…”

Section: Introductionmentioning

confidence: 99%

A New Projection Algorithm for Solving a System of Nonlinear Equations With Convex Constraints

Lian¹

2013

Bulletin of the Korean Mathematical Society

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Abstract. We present a new algorithm for solving a system of nonlinear equations with convex constraints which combines proximal point and projection methodologies. Compared with the existing projection methods for solving the problem, we use a different system of linear equations to obtain the proximal point; and moreover, at the step of getting next iterate, our projection way and projection region are also different. Based on the Armijo-type line search procedure, a new hyperplane is introduced. Using the separate property of hyperplane, the new algorithm is proved to be globally convergent under much weaker assumptions than monotone or more generally pseudomonotone. We study the convergence rate of the iterative sequence under very mild error bound conditions.

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“…Generally, the quasi-Newton direction is not a descent direction for the merit function, which makes it difficult to globalize the method. In this paper, based on the hyperplane projection method [23], we propose a BFGS method for solving nonlinear monotone equations and prove its global convergence property without use of merit functions. The differentiability of the equation is also not assumed.…”

Section: 1)mentioning

confidence: 99%

“…First we recall the hyperplane projection method [23] for solving nonlinear monotone equations (1.1). By the monotonicity of F , we have…”

Section: Algorithmmentioning

confidence: 99%

A globally convergent BFGS method for nonlinear monotone equations without any merit functions

Zhou

2008

Math. Comp.

116

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Abstract. Since 1965, there has been significant progress in the theoretical study on quasi-Newton methods for solving nonlinear equations, especially in the local convergence analysis. However, the study on global convergence of quasi-Newton methods is relatively fewer, especially for the BFGS method. To ensure global convergence, some merit function such as the squared norm merit function is typically used. In this paper, we propose an algorithm for solving nonlinear monotone equations, which combines the BFGS method and the hyperplane projection method. We also prove that the proposed BFGS method converges globally if the equation is monotone and Lipschitz continuous without differentiability requirement on the equation, which makes it possible to solve some nonsmooth equations. An attractive property of the proposed method is that its global convergence is independent of any merit function.We also report some numerical results to show efficiency of the proposed method.

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A Globally Convergent Inexact Newton Method for Systems of Monotone Equations

Cited by 223 publications

References 12 publications

Forcing strong convergence of proximal point iterations in a Hilbert space

Forcing strong convergence of proximal point iterations in a Hilbert space

A New Projection Algorithm for Solving a System of Nonlinear Equations With Convex Constraints

A globally convergent BFGS method for nonlinear monotone equations without any merit functions

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