Unconstrained optimization reformulations of variational inequality problems.

*(English)*Zbl 0879.90180Summary: Recently, J. M. Peng [‘Equivalence of variational inequality problems to unconstrained optimization’, Tech. Report, State Key Lab. of Sci. and Eng. Computing, Academia Sinica, Beijing (1995)] considered a merit function for the variational inequality problem (VIP), which constitutes an unconstrained differentiable optimization reformulation of VIP. We generalize the merit function proposed by Peng and study various properties of the generalized function. We call this function the \(D\)-gap function. We give conditions under which any stationary point of the \(D\)-gap function is a solution of VIP and conditions under which it provides a global error bound for VIP. We also present a descent method for solving VIP based on the \(D\)-gap function.

##### MSC:

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

49J40 | Variational inequalities |

##### Keywords:

merit function; variational inequality; \(D\)-gap function; global error bound; descent method
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\textit{N. Yamashita} et al., J. Optim. Theory Appl. 92, No. 3, 439--456 (1997; Zbl 0879.90180)

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##### References:

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