Clarke directional derivatives of regularized gap functions for nonsmooth quasi-variational inequalities

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September 2014, 4(3): 365-379. doi: 10.3934/mcrf.2014.4.365

Clarke directional derivatives of regularized gap functions for nonsmooth quasi-variational inequalities

Haisen Zhang ^1,

1.	School of Mathematics, Sichuan University, 610064, Chengdu, China

Received July 2013 Revised November 2013 Published April 2014

Abstract
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In optimization problems, it is significant to study the directional derivatives and subdifferentials of objective functions. Using directional derivatives and subdifferentials of objective functions, we can establish optimality conditions, derive error bound properties, and propose optimal algorithms. In this paper, the upper and lower estimates for the Clarke directional derivatives of a class of marginal functions are established. Employing this result, we obtain the exact formulations of the Clarke directional derivatives of the regularized gap functions for nonsmooth quasi-variational inequalities.

Keywords: Nonsmooth quasi-variational inequality, Clarke directional derivative., regularized gap function.

Mathematics Subject Classification: Primary: 90C33; Secondary: 49J52, 49J53, 90C2.

Citation: Haisen Zhang. Clarke directional derivatives of regularized gap functions for nonsmooth quasi-variational inequalities. Mathematical Control and Related Fields, 2014, 4 (3) : 365-379. doi: 10.3934/mcrf.2014.4.365

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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