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**Convergence and error bound of a method for solving variational inequality problems via the generalized D-gap function.**
*(English)*
Zbl 1045.49015

Summary: The Variational Inequality Problem (VIP) can be reformulated as an unconstrained minimization problem through the generalized D-gap function. Recently, a hybrid Newton-type method was proposed by J. M. Peng and M. Fukushima [J. Math. Program. 86 A, No. 2, 367–386 (1999; Zbl 0939.90023)] for minimizing a special form of the generalized D-gap function. In this paper, the hybrid Newton-type algorithm is extended to minimize the general form \(g_{\alpha\beta}\) of the generalized D-gap function. It is shown that the algorithm has nice convergence properties. Under some reasonable conditions, it is proved that the algorithm is locally and globally convergent. Moreover, it is proved that the function \(g_{\alpha\beta}\) has bounded level sets for strongly monotone VIP. An error bound of the algorithm is obtained.

### MSC:

49J40 | Variational inequalities |

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

47J25 | Iterative procedures involving nonlinear operators |

### Keywords:

variational inequality problems; unconstrained optimization; generalized D-gap function; global convergence; error bound estimation### Citations:

Zbl 0939.90023
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\textit{B. Qu} et al., J. Optim. Theory Appl. 119, No. 3, 535--552 (2003; Zbl 1045.49015)

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### References:

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