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

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 & 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. doi: 10.1007/bfb0120841. Google Scholar
[2]	D. P. Bertsekas, A. Nedic and A. E. Ozdaglar, Convex Analysis and Optimization,, Athena Scientific, (2003). Google Scholar
[3]	D. Chan and J. S. Pang, The generalized quasi-variational inequality problem,, Math. Oper. Res., 7 (1982), 211. doi: 10.1287/moor.7.2.211. Google Scholar
[4]	F. H. Clarke, Optimization and Nonsmooth Analysis,, Wiley, (1983). doi: 10.1137/1.9781611971309. Google Scholar
[5]	F. Facchi and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems,, Vol. I, (2003). doi: 10.1007/b97543. Google Scholar
[6]	M. Fukushima, Equivalent differentiable optimaization problems and descent methods for asymmetric variational inequality problems,, Math. Program., 53 (1992), 99. doi: 10.1007/BF01585696. Google Scholar
[7]	M. Fukushima, A class of gap functions for quasi-variational inequality problems,, J. Indust. Manage. Optim., 3 (2007), 165. doi: 10.3934/jimo.2007.3.165. Google Scholar
[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. doi: 10.1080/10556788.2013.841694. Google Scholar
[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. doi: 10.1007/bfb0066253. Google Scholar
[10]	W. W. Hogan, Point-to-set maps in mathematical programming,, SIAM review, 15 (1973), 591. doi: 10.1137/1015073. Google Scholar
[11]	W. W. Hogan, Directional derivatives for extremal-value functions with applications to the completely convex case,, Oper. Res., 21 (1973), 188. doi: 10.1287/opre.21.1.188. Google Scholar
[12]	K. Kubota and M. Fukushima, Gap function approach to the generalized Nash equilibrium problem,, J. Optim. Theory Appl., 144* (2010), 511. doi: 10.1007/s10957-009-9614-4. Google Scholar*
[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. doi: 10.1137/070696283. Google Scholar
[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. doi: 10.1007/BF02189796. Google Scholar
[15]	B. S. Mordukhovich, N. M. Nam and N. D. Yen, Subgradients of marginal functions in parametric mathematical programming,, Math. Program., 116* (2009), 369. doi: 10.1007/s10107-007-0120-x. Google Scholar*
[16]	K. F. Ng and L. L. Tan, Error bounds of regularized gap functions for nonsmooth variational inequality problems,, Math. Program., 110* (2007), 405. doi: 10.1007/s10107-006-0007-2. Google Scholar*
[17]	K. F. Ng and L. L. Tan, D-Gap Functions for nonsmooth variational inequality problems,, J. Optim. Theory Appl., 133* (2007), 77. doi: 10.1007/s10957-007-9193-1. Google Scholar*
[18]	J.-M. Peng, Equivalence of variational inequality problems to unconstrained minimization,, Math. program., 78 (1997), 347. doi: 10.1007/BF02614360. Google Scholar
[19]	R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming,, Math. Program. Study, 17 (1982), 28. doi: 10.1007/bfb0120958. Google Scholar
[20]	E. M. Stern, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970). Google Scholar
[21]	K. Taji, On gap functions for quasi-variational inequalities,, Abstract Appl. Anal., 2008* (2008). doi: 10.1155/2008/531361. Google Scholar*
[22]	L. L. Tan, Regularized gap functions for nonsmooth variational inequality problems,, J. Math. Anal. Appl., 334* (2007), 1022. doi: 10.1016/j.jmaa.2007.01.025. Google Scholar*
[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. doi: 10.1023/A:1010953431290. Google Scholar*
[24]	N. Yamashita, K. Taji and M. Fukushima, Unconstrained optimization reformulations of asymmetric variational inequality problems,, J. Optim. Theory Appl., 92 (1997), 439. doi: 10.1023/A:1022660704427. Google Scholar

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References:

[1]	A. Auslender, Differential stability in nonconvex and nondifferentiable programming,, Math. Program. Study, 10 (1979), 29. doi: 10.1007/bfb0120841. Google Scholar
[2]	D. P. Bertsekas, A. Nedic and A. E. Ozdaglar, Convex Analysis and Optimization,, Athena Scientific, (2003). Google Scholar
[3]	D. Chan and J. S. Pang, The generalized quasi-variational inequality problem,, Math. Oper. Res., 7 (1982), 211. doi: 10.1287/moor.7.2.211. Google Scholar
[4]	F. H. Clarke, Optimization and Nonsmooth Analysis,, Wiley, (1983). doi: 10.1137/1.9781611971309. Google Scholar
[5]	F. Facchi and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems,, Vol. I, (2003). doi: 10.1007/b97543. Google Scholar
[6]	M. Fukushima, Equivalent differentiable optimaization problems and descent methods for asymmetric variational inequality problems,, Math. Program., 53 (1992), 99. doi: 10.1007/BF01585696. Google Scholar
[7]	M. Fukushima, A class of gap functions for quasi-variational inequality problems,, J. Indust. Manage. Optim., 3 (2007), 165. doi: 10.3934/jimo.2007.3.165. Google Scholar
[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. doi: 10.1080/10556788.2013.841694. Google Scholar
[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. doi: 10.1007/bfb0066253. Google Scholar
[10]	W. W. Hogan, Point-to-set maps in mathematical programming,, SIAM review, 15 (1973), 591. doi: 10.1137/1015073. Google Scholar
[11]	W. W. Hogan, Directional derivatives for extremal-value functions with applications to the completely convex case,, Oper. Res., 21 (1973), 188. doi: 10.1287/opre.21.1.188. Google Scholar
[12]	K. Kubota and M. Fukushima, Gap function approach to the generalized Nash equilibrium problem,, J. Optim. Theory Appl., 144* (2010), 511. doi: 10.1007/s10957-009-9614-4. Google Scholar*
[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. doi: 10.1137/070696283. Google Scholar
[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. doi: 10.1007/BF02189796. Google Scholar
[15]	B. S. Mordukhovich, N. M. Nam and N. D. Yen, Subgradients of marginal functions in parametric mathematical programming,, Math. Program., 116* (2009), 369. doi: 10.1007/s10107-007-0120-x. Google Scholar*
[16]	K. F. Ng and L. L. Tan, Error bounds of regularized gap functions for nonsmooth variational inequality problems,, Math. Program., 110* (2007), 405. doi: 10.1007/s10107-006-0007-2. Google Scholar*
[17]	K. F. Ng and L. L. Tan, D-Gap Functions for nonsmooth variational inequality problems,, J. Optim. Theory Appl., 133* (2007), 77. doi: 10.1007/s10957-007-9193-1. Google Scholar*
[18]	J.-M. Peng, Equivalence of variational inequality problems to unconstrained minimization,, Math. program., 78 (1997), 347. doi: 10.1007/BF02614360. Google Scholar
[19]	R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming,, Math. Program. Study, 17 (1982), 28. doi: 10.1007/bfb0120958. Google Scholar
[20]	E. M. Stern, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970). Google Scholar
[21]	K. Taji, On gap functions for quasi-variational inequalities,, Abstract Appl. Anal., 2008* (2008). doi: 10.1155/2008/531361. Google Scholar*
[22]	L. L. Tan, Regularized gap functions for nonsmooth variational inequality problems,, J. Math. Anal. Appl., 334* (2007), 1022. doi: 10.1016/j.jmaa.2007.01.025. Google Scholar*
[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. doi: 10.1023/A:1010953431290. Google Scholar*
[24]	N. Yamashita, K. Taji and M. Fukushima, Unconstrained optimization reformulations of asymmetric variational inequality problems,, J. Optim. Theory Appl., 92 (1997), 439. doi: 10.1023/A:1022660704427. Google Scholar

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