AUTHORS: Zili Wu
Download as PDF
ABSTRACT: For a variational inequality problem (VIP) with a psudomonotone mapping F on its solution set C*, we give equivalent statements for C* to be determined by the zeroes Γ(c*) of the primal gap function of VIP, where c*є C*. One sufficient condition is also presented in terms of weaker sharpness of C*. With the psudomonotonicity* of F on C being characterized, C* turns out to coincide with the zeroes Λ(c*) of the dual gap function of VIP. If also F has the same direction on Γ(c*), then Γ(c*) coincides with C*, Λ(c*), and the solution set C* of the dual variational inequality problem. This has further been shown to be equivalent to saying that F is constant on Γ(c*) when F is psudomonotonone+ on C*.
KEYWORDS: Variational inequality, minimum principle sufficiency, weaker sharpness, pseudomonotonicity*, gap functions
REFERENCES:
[1] J. V. Burke and M. C. Ferris, Weak Sharp Minimum in Mathematical Programming, SIAM J. Control and Optimization 31, 1993, pp. 1340- 1359.
[2] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York 1983; reprinted as vol. 5 of Classics in Applied Mathematics, SIAM, Philadelphia, PA, 1990.
[3] M. C. Ferris and O. L. Mangasarian, Minimum principle sufficiency, Mathematical Programming 57, 1992, pp. 1-14.
[4] F. Facchinei and J .S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, 2 volumes, Springer, 2003.
[5] P. T. Harker and J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Program. 48, 1990, pp. 161-220.
[6] Y. H. Hu and W. Song, Weak sharp solutions for variational inequalities in Banach spaces, J. Math. Anal. Appl. 374, 2011, pp. 118-132.
[7] Y. N. Liu and Z. L. Wu, Characterization of weakly sharp solutions of a variational inequality by its primal gap function, Optimization Letters 10, 2016, pp. 563-576.
[8] P. Marcotte and D. L. Zhu, Weak sharp solutions of variational inequalities, SIAM J. Optim. 9, 1998, pp. 179-189.
[9] M. Patriksson, A unified framework of descent algorithms for nonlinear programs and variational inequalities, Ph. D. thesis, Department of Mathematics, Linko¨ping Institute of Technology, Linko¨ping, Sweden, 1993.
[10] Z. L. Wu and S. Y. Wu, Weak sharp solutions of variational inequalities in Hilbert spaces, SIAM J. Optim. 14, 2004, pp. 1011-1027.
[11] Z. L. Wu and S. Y. Wu, Gateaux differentiability ˆ of the dual gap function of a variational inequality, European J. Oper. Res. 190, 2008, pp. 328- 344.
[12] Z. L. Wu, Characterizations of Weak Sharp Solutions for a Variational Inequality with a Psudomonotone Mapping, submitted for publication.
[13] J. Z. Zhang, C. Y. Wan and N. H. Xiu, The dual gap functions for variational inequalities, Appl. Math. Optim. 48, 2003, pp. 129-148.