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Clarke directional derivatives of regularized gap functions for nonsmooth quasi-variational inequalities
1. | School of Mathematics, Sichuan University, 610064, Chengdu, China |
References:
[1] |
A. Auslender, Differential stability in nonconvex and nondifferentiable programming, Math. Program. Study, 10 (1979), 29-41.
doi: 10.1007/bfb0120841. |
[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.
doi: 10.1287/moor.7.2.211. |
[4] |
F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
doi: 10.1137/1.9781611971309. |
[5] |
F. Facchi and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. I, Springer-Verlag, New York, 2003.
doi: 10.1007/b97543. |
[6] |
M. Fukushima, Equivalent differentiable optimaization problems and descent methods for asymmetric variational inequality problems, Math. Program., 53 (1992), 99-110.
doi: 10.1007/BF01585696. |
[7] |
M. Fukushima, A class of gap functions for quasi-variational inequality problems, J. Indust. Manage. Optim., 3 (2007), 165-171.
doi: 10.3934/jimo.2007.3.165. |
[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.
doi: 10.1080/10556788.2013.841694. |
[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.
doi: 10.1007/bfb0066253. |
[10] |
W. W. Hogan, Point-to-set maps in mathematical programming, SIAM review, 15 (1973), 591-603.
doi: 10.1137/1015073. |
[11] |
W. W. Hogan, Directional derivatives for extremal-value functions with applications to the completely convex case, Oper. Res., 21 (1973), 188-209.
doi: 10.1287/opre.21.1.188. |
[12] |
K. Kubota and M. Fukushima, Gap function approach to the generalized Nash equilibrium problem, J. Optim. Theory Appl., 144 (2010), 511-531.
doi: 10.1007/s10957-009-9614-4. |
[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.
doi: 10.1137/070696283. |
[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.
doi: 10.1007/BF02189796. |
[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.
doi: 10.1007/s10107-007-0120-x. |
[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.
doi: 10.1007/s10107-006-0007-2. |
[17] |
K. F. Ng and L. L. Tan, D-Gap Functions for nonsmooth variational inequality problems, J. Optim. Theory Appl., 133 (2007), 77-97.
doi: 10.1007/s10957-007-9193-1. |
[18] |
J.-M. Peng, Equivalence of variational inequality problems to unconstrained minimization, Math. program., 78 (1997), 347-355.
doi: 10.1007/BF02614360. |
[19] |
R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, Math. Program. Study, 17 (1982), 28-66.
doi: 10.1007/bfb0120958. |
[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.
doi: 10.1155/2008/531361. |
[22] |
L. L. Tan, Regularized gap functions for nonsmooth variational inequality problems, J. Math. Anal. Appl., 334 (2007), 1022-1038.
doi: 10.1016/j.jmaa.2007.01.025. |
[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.
doi: 10.1023/A:1010953431290. |
[24] |
N. Yamashita, K. Taji and M. Fukushima, Unconstrained optimization reformulations of asymmetric variational inequality problems, J. Optim. Theory Appl., 92 (1997), 439-456.
doi: 10.1023/A:1022660704427. |
show all references
References:
[1] |
A. Auslender, Differential stability in nonconvex and nondifferentiable programming, Math. Program. Study, 10 (1979), 29-41.
doi: 10.1007/bfb0120841. |
[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.
doi: 10.1287/moor.7.2.211. |
[4] |
F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
doi: 10.1137/1.9781611971309. |
[5] |
F. Facchi and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. I, Springer-Verlag, New York, 2003.
doi: 10.1007/b97543. |
[6] |
M. Fukushima, Equivalent differentiable optimaization problems and descent methods for asymmetric variational inequality problems, Math. Program., 53 (1992), 99-110.
doi: 10.1007/BF01585696. |
[7] |
M. Fukushima, A class of gap functions for quasi-variational inequality problems, J. Indust. Manage. Optim., 3 (2007), 165-171.
doi: 10.3934/jimo.2007.3.165. |
[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.
doi: 10.1080/10556788.2013.841694. |
[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.
doi: 10.1007/bfb0066253. |
[10] |
W. W. Hogan, Point-to-set maps in mathematical programming, SIAM review, 15 (1973), 591-603.
doi: 10.1137/1015073. |
[11] |
W. W. Hogan, Directional derivatives for extremal-value functions with applications to the completely convex case, Oper. Res., 21 (1973), 188-209.
doi: 10.1287/opre.21.1.188. |
[12] |
K. Kubota and M. Fukushima, Gap function approach to the generalized Nash equilibrium problem, J. Optim. Theory Appl., 144 (2010), 511-531.
doi: 10.1007/s10957-009-9614-4. |
[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.
doi: 10.1137/070696283. |
[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.
doi: 10.1007/BF02189796. |
[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.
doi: 10.1007/s10107-007-0120-x. |
[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.
doi: 10.1007/s10107-006-0007-2. |
[17] |
K. F. Ng and L. L. Tan, D-Gap Functions for nonsmooth variational inequality problems, J. Optim. Theory Appl., 133 (2007), 77-97.
doi: 10.1007/s10957-007-9193-1. |
[18] |
J.-M. Peng, Equivalence of variational inequality problems to unconstrained minimization, Math. program., 78 (1997), 347-355.
doi: 10.1007/BF02614360. |
[19] |
R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, Math. Program. Study, 17 (1982), 28-66.
doi: 10.1007/bfb0120958. |
[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.
doi: 10.1155/2008/531361. |
[22] |
L. L. Tan, Regularized gap functions for nonsmooth variational inequality problems, J. Math. Anal. Appl., 334 (2007), 1022-1038.
doi: 10.1016/j.jmaa.2007.01.025. |
[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.
doi: 10.1023/A:1010953431290. |
[24] |
N. Yamashita, K. Taji and M. Fukushima, Unconstrained optimization reformulations of asymmetric variational inequality problems, J. Optim. Theory Appl., 92 (1997), 439-456.
doi: 10.1023/A:1022660704427. |
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