Abstract
In this paper, we investigate the regularized gap function due to Auchmuty (1989) and Fukushima (1992), and the D-gap function due to Peng (1997) for a nonsmooth variational inequality problem where the mapping involved is Lipschitz continuous but not necessarily continuously differentiable. Under the semismoothness and strong monotonicity, we prove that the solutions of the nonsmooth variational inequality problem coincide with the stationary points of the gap functions. A descent algorithm is proposed for finding a stationary point of the D-gap function. The algorithm differs from the usual nonsmooth descent algorithms in that at each step the computation of a descent direction involves merely the function value. In order to improve the convergence locally, a generalized Newton’s algorithm is proposed. Finally, initial numerical results are reported.
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Xu, H. (2001). Regularized Gap Functions and D-Gap Functions for Nonsmooth Variational Inequalities. In: Rubinov, A., Glover, B. (eds) Optimization and Related Topics. Applied Optimization, vol 47. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6099-6_11
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DOI: https://doi.org/10.1007/978-1-4757-6099-6_11
Publisher Name: Springer, Boston, MA
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