Mathematical Control and Related Fields (MCRF)

Clarke directional derivatives of regularized gap functions for nonsmooth quasi-variational inequalities
Pages: 365 - 379, Issue 3, September 2014

doi:10.3934/mcrf.2014.4.365      Abstract        References        Full text (358.2K)           Related Articles

Haisen Zhang - School of Mathematics, Sichuan University, 610064, Chengdu, China (email)

1 A. Auslender, Differential stability in nonconvex and nondifferentiable programming, Math. Program. Study, 10 (1979), 29-41.       
2 D. P. Bertsekas, A. Nedic and A. E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, Belmont, 2003.       
3 D. Chan and J. S. Pang, The generalized quasi-variational inequality problem, Math. Oper. Res., 7 (1982), 211-222.       
4 F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.       
5 F. Facchi and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. I, Springer-Verlag, New York, 2003.       
6 M. Fukushima, Equivalent differentiable optimaization problems and descent methods for asymmetric variational inequality problems, Math. Program., 53 (1992), 99-110.       
7 M. Fukushima, A class of gap functions for quasi-variational inequality problems, J. Indust. Manage. Optim., 3 (2007), 165-171.       
8 N. Harms, C. Kanzow and O. Stein, Smoothness properties of a regularized gap function for quasi-variational inequalities, Optim. Methods Softw., 29 (2014), 720-750.       
9 J. -B. Hiriart-Urruty, Approximate first-Order and second-order directional derivatives of a marginal function in convex optimization, J. Optim. Theory Appl., 48 (1986), 127-140.       
10 W. W. Hogan, Point-to-set maps in mathematical programming, SIAM review, 15 (1973), 591-603.       
11 W. W. Hogan, Directional derivatives for extremal-value functions with applications to the completely convex case, Oper. Res., 21 (1973), 188-209.       
12 K. Kubota and M. Fukushima, Gap function approach to the generalized Nash equilibrium problem, J. Optim. Theory Appl., 144 (2010), 511-531.       
13 G. Li and K. F. Ng, Error bounds of generalized D-gap functions for nonsmooth and nonmonotone variational inequality problems, SIAM J. Optim., 20 (2009), 667-690.       
14 L. I. Minchenko and P. P. Sakolchik, Hölder behavior of optimal solutions and directional differentiability of marginal functions in nonlinear programming, J. Optim. Theory Appl., 90 (1996), 555-580.       
15 B. S. Mordukhovich, N. M. Nam and N. D. Yen, Subgradients of marginal functions in parametric mathematical programming, Math. Program., 116 (2009), 369-396.       
16 K. F. Ng and L. L. Tan, Error bounds of regularized gap functions for nonsmooth variational inequality problems, Math. Program., 110 (2007), 405-429.       
17 K. F. Ng and L. L. Tan, D-Gap Functions for nonsmooth variational inequality problems, J. Optim. Theory Appl., 133 (2007), 77-97.       
18 J.-M. Peng, Equivalence of variational inequality problems to unconstrained minimization, Math. program., 78 (1997), 347-355.       
19 R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, Math. Program. Study, 17 (1982), 28-66.       
20 E. M. Stern, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.       
21 K. Taji, On gap functions for quasi-variational inequalities, Abstract Appl. Anal., 2008 (2008), Art. ID 531361, 7 pages.       
22 L. L. Tan, Regularized gap functions for nonsmooth variational inequality problems, J. Math. Anal. Appl., 334 (2007), 1022-1038.       
23 D. E. Ward and G. M. Lee, Upper subderivatives and generalized gradients of the marginal function of a non-Lipschitzian program, Ann. Oper. Res., 101 (2001), 299-312.       
24 N. Yamashita, K. Taji and M. Fukushima, Unconstrained optimization reformulations of asymmetric variational inequality problems, J. Optim. Theory Appl., 92 (1997), 439-456.       

Go to top