Solving nonlinear complementarity problems by isotonicity of the metric projection

Abbas, Mujahid; Németh, S. Z.

doi:10.1016/j.jmaa.2011.08.048

Cited by 21 publications

(43 citation statements)

References 16 publications

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“…Then the assumption T −1 < 1 in Proposition 1 is sufficient to the uniqueness of solution of (1). The next example shows that it is not possible to increase the upper bound of T −1 and still ensure the uniqueness of solution in (1).…”

Section: A Semi-smooth Newton Methods For a Piecewise Linear Systemsmentioning

confidence: 97%

“…In this section we present and analyze the semi-smooth Newton's method for solving (1). We begin with an existence result of solution to the equation (1).…”

Section: A Semi-smooth Newton Methods For a Piecewise Linear Systemsmentioning

confidence: 99%

“…Note that T −1 = 1 and there holds F (x * ) = F (x * * ) = 0, where x * = [1,1] T and x * * = [0, 1] T .…”

Section: A Semi-smooth Newton Methods For a Piecewise Linear Systemsmentioning

confidence: 99%

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A semi-smooth Newton method for a special piecewise linear system with application to positively constrained convex quadratic programming

Barrios¹,

Bello-Cruz²,

Ferreira³

et al. 2016

Journal of Computational and Applied Mathematics

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In this paper a special piecewise linear system is studied. It is shown that, under a mild assumption, the semi-smooth Newton method applied to this system is well defined and the method generates a sequence that converges linearly to a solution. Besides, we also show that the generated sequence is bounded, for any starting point, and a formula for any accumulation point of this sequence is presented. As an application, we study the convex quadratic programming problem under positive constraints. The numerical results suggest that the semi-smooth Newton method achieves accurate solutions to large scale problems in few iterations.

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Section: A Semi-smooth Newton Methods For a Piecewise Linear Systemsmentioning

confidence: 97%

“…In this section we present and analyze the semi-smooth Newton's method for solving (1). We begin with an existence result of solution to the equation (1).…”

Section: A Semi-smooth Newton Methods For a Piecewise Linear Systemsmentioning

confidence: 99%

See 1 more Smart Citation

A semi-smooth Newton method for a special piecewise linear system with application to positively constrained convex quadratic programming

Barrios¹,

Bello-Cruz²,

Ferreira³

et al. 2016

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Therefore, as C < 1 we may apply Theorem 1 with X = R m , T = F , d(x, y) = y − x for all x, y ∈ R m and α = C , for concluding that the Picard's Method (13) or equivalently, the sequence…”

Section: Picard's Methods For Solving Problemmentioning

confidence: 99%

Projection onto simplicial cones by Picard's method

Barrios¹,

Ferreira²,

Németh

2015

Linear Algebra and its Applications

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By using Moreau's decomposition theorem for projecting onto cones, the problem of projecting onto a simplicial cone is reduced to finding the unique solution of a nonsmooth system of equations. It is shown that Picard's method applied to the system of equations associated to the problem of projecting onto a simplicial cone generates a sequence that converges linearly to the solution of the system. Numerical experiments are presented making the comparison between Picard's and semi-smooth Newton's methods to solve the nonsmooth system associated with the problem of projecting a point onto a simplicial cone.

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“…If F is continuous and {x n } n∈N is convergent to x * , then by a simple limiting process in (5), it follows that x * is a fixed point of the mapping P K • (I − F ) and hence a solution of V I(F, K). Therefore, it is natural to seek convergence conditions for x n .…”

Section: Variational Inequalitiesmentioning

confidence: 99%

Extended Lorentz Cones and Variational Inequalities on Cylinders

Németh

Zhang

2015

J Optim Theory Appl

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Solutions of a variational inequality problem defined by a closed and convex set and a mapping are found by imposing conditions for the monotone convergence with respect to a cone of the Picard iteration corresponding to the composition of the projection onto the defining closed and convex set and the difference in the identity mapping and the defining mapping. One of these conditions is the isotonicity of the projection onto the defining closed and convex set. If the closed and convex set is a cylinder and the cone is an extented Lorentz cone, then this condition can be dropped because it is automatically satisfied. In this case, a large class of affine mappings and cylinders which satisfy the conditions of monotone convergence above is presented. The obtained results are further specialized for unbounded box-constrained variational inequalities. In a particular case of a cylinder with a base being a cone, the variational inequality is reduced to a generalized mixed complementarity problem which has been already considered in Németh and Zhang (J Global Optim 62(3): [443][444][445][446][447][448][449][450][451][452][453][454][455][456][457] 2015).

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Solving nonlinear complementarity problems by isotonicity of the metric projection

Cited by 21 publications

References 16 publications

A semi-smooth Newton method for a special piecewise linear system with application to positively constrained convex quadratic programming

A semi-smooth Newton method for a special piecewise linear system with application to positively constrained convex quadratic programming

Projection onto simplicial cones by Picard's method

Extended Lorentz Cones and Variational Inequalities on Cylinders

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