A globally convergent Newton method for solving strongly monotone variational inequalities

Taji, Kouichi; Fukushima, Masao; Ibaraki, Toshihide

doi:10.1007/bf01581276

Cited by 169 publications

(53 citation statements)

References 16 publications

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“…To our knowledge, the only merit functions which have been used to globalize the Newton method of the form (1.2), (1.3) are the gap function [25], the regularized gap function [51,50], the D-gap function [31,32], and (the square of) the FischerBurmeister function [8]. (We note that some of these methods were developed for the more general mixed complementarity or variational inequality setting.)…”

Section: Introductionmentioning

confidence: 99%

“…Using the regularized gap function [51,50] admits inexact Armijo-type linesearch but requires strong monotonicity of F for global convergence (see also [52]). In addition, methods of [25,51,50] also need the (restrictive) strict complementarity assumption to establish superlinear/quadratic rate of convergence. Note that the subproblems in [25,51,50] are solvable due to the compactness of the feasible set and the strong monotonicity of F , respectively.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, methods of [25,51,50] also need the (restrictive) strict complementarity assumption to establish superlinear/quadratic rate of convergence. Note that the subproblems in [25,51,50] are solvable due to the compactness of the feasible set and the strong monotonicity of F , respectively.…”

Section: Introductionmentioning

confidence: 99%

“…This feature of approximate subproblem solutions is of particular importance for large-scale problems. (We note that the effect of an inexact subproblem solution has not been analyzed in globalizations proposed in [51,50,31,32,8].) Under the assumptions of positive definiteness of the Jacobian of F at the solution, and its Hölder continuity in some neighborhood of it, the local superlinear rate of convergence is also established (see Theorem 4.3).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Truly Globally Convergent Newton-Type Method for the Monotone Nonlinear Complementarity Problem

Solodov¹,

Svaiter²

2000

SIAM J. Optim.

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Abstract. The Josephy-Newton method for solving a nonlinear complementarity problem consists of solving, possibly inexactly, a sequence of linear complementarity problems. Under appropriate regularity assumptions, this method is known to be locally (superlinearly) convergent. To enlarge the domain of convergence of the Newton method, some globalization strategy based on a chosen merit function is typically used. However, to ensure global convergence to a solution, some additional restrictive assumptions are needed. These assumptions imply boundedness of level sets of the merit function and often even (global) uniqueness of the solution. We present a new globalization strategy for monotone problems which is not based on any merit function. Our linesearch procedure utilizes the regularized Newton direction and the monotonicity structure of the problem to force global convergence by means of a (computationally explicit) projection step which reduces the distance to the solution set of the problem. The resulting algorithm is truly globally convergent in the sense that the subproblems are always solvable, and the whole sequence of iterates converges to a solution of the problem without any regularity assumptions. In fact, the solution set can even be unbounded. Each iteration of the new method has the same order of computational cost as an iteration of the damped Newton method. Under natural assumptions, the local superlinear rate of convergence is also achieved.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Truly Globally Convergent Newton-Type Method for the Monotone Nonlinear Complementarity Problem

Solodov¹,

Svaiter²

2000

SIAM J. Optim.

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“…and that the VIP is reformulated as the equivalent nonlinear optimization problem by using the gap function, the regularized gap function [6,13,16] and so on. Continuation methods are one approach to solve the system of nonlinear equations.…”

mentioning

confidence: 99%

A Note on Continuation Method for Strongly Monotone Variational Inequalities under Slater's Constraint Qualification

Taji¹,

Uchiyama²

2003

Transactions of the Institute of Systems, Control and Informati

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Variational Inequalities

Konnov¹

2011

Wiley Encyclopedia of Operations Research and Management Science

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We consider variational inequality problems in a finite‐dimensional space setting. We first describe existence and uniqueness results under norm monotonicity assumptions. Relations to some other general nonlinear problems are discussed. Then we describe basic solution methods utilizing the above monotonicity concepts and their convergence properties. Some extensions for the multivalued case are also discussed.

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A globally convergent Newton method for solving strongly monotone variational inequalities

Cited by 169 publications

References 16 publications

A Truly Globally Convergent Newton-Type Method for the Monotone Nonlinear Complementarity Problem

A Truly Globally Convergent Newton-Type Method for the Monotone Nonlinear Complementarity Problem

A Note on Continuation Method for Strongly Monotone Variational Inequalities under Slater's Constraint Qualification

Variational Inequalities

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