American Institute of Mathematical Sciences

Advanced
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

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

show all references

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

[1]

Masao Fukushima. A class of gap functions for quasi-variational inequality problems. Journal of Industrial & Management Optimization, 2007, 3 (2) : 165-171. doi: 10.3934/jimo.2007.3.165
[2]

Takeshi Fukao, Nobuyuki Kenmochi. Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2523-2538. doi: 10.3934/dcds.2015.35.2523
[3]

Suxiang He, Pan Zhang, Xiao Hu, Rong Hu. A sample average approximation method based on a D-gap function for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 977-987. doi: 10.3934/jimo.2014.10.977
[4]

Lori Badea. Multigrid methods for some quasi-variational inequalities. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1457-1471. doi: 10.3934/dcdss.2013.6.1457
[5]

Yusuke Murase, Risei Kano, Nobuyuki Kenmochi. Elliptic Quasi-variational inequalities and applications. Conference Publications, 2009, 2009 (Special) : 583-591. doi: 10.3934/proc.2009.2009.583
[6]

Yurii Nesterov, Laura Scrimali. Solving strongly monotone variational and quasi-variational inequalities. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1383-1396. doi: 10.3934/dcds.2011.31.1383
[7]

Nobuyuki Kenmochi. Parabolic quasi-variational diffusion problems with gradient constraints. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 423-438. doi: 10.3934/dcdss.2013.6.423
[8]

Laura Scrimali. Mixed behavior network equilibria and quasi-variational inequalities. Journal of Industrial & Management Optimization, 2009, 5 (2) : 363-379. doi: 10.3934/jimo.2009.5.363
[9]

Yusuke Murase, Atsushi Kadoya, Nobuyuki Kenmochi. Optimal control problems for quasi-variational inequalities and its numerical approximation. Conference Publications, 2011, 2011 (Special) : 1101-1110. doi: 10.3934/proc.2011.2011.1101
[10]

Samir Adly, Tahar Haddad. On evolution quasi-variational inequalities and implicit state-dependent sweeping processes. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020105
[11]

Minghua Li, Chunrong Chen, Shengjie Li. Error bounds of regularized gap functions for nonmonotone Ky Fan inequalities. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2019001
[12]

A. M. Bagirov, Moumita Ghosh, Dean Webb. A derivative-free method for linearly constrained nonsmooth optimization. Journal of Industrial & Management Optimization, 2006, 2 (3) : 319-338. doi: 10.3934/jimo.2006.2.319
[13]

Dorina Mitrea, Marius Mitrea, Sylvie Monniaux. The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1295-1333. doi: 10.3934/cpaa.2008.7.1295
[14]

Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709
[15]

Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437
[16]

Carlo Sinestrari. Semiconcavity of the value function for exit time problems with nonsmooth target. Communications on Pure & Applied Analysis, 2004, 3 (4) : 757-774. doi: 10.3934/cpaa.2004.3.757
[17]

S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial & Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155
[18]

Junkee Jeon, Jehan Oh. Valuation of American strangle option: Variational inequality approach. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 755-781. doi: 10.3934/dcdsb.2018206
[19]

Meina Gao, Jianjun Liu. Quasi-periodic solutions for derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101
[20]

Fangfang Dong, Yunmei Chen. A fractional-order derivative based variational framework for image denoising. Inverse Problems & Imaging, 2016, 10 (1) : 27-50. doi: 10.3934/ipi.2016.10.27

2018 Impact Factor: 1.292

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences