# Revisiting the construction of gap functions for variational inequalities and equilibrium problems via conjugate duality

Open Mathematics (2013)

- Volume: 11, Issue: 5, page 829-850
- ISSN: 2391-5455

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topLiana Cioban, and Ernö Csetnek. "Revisiting the construction of gap functions for variational inequalities and equilibrium problems via conjugate duality." Open Mathematics 11.5 (2013): 829-850. <http://eudml.org/doc/269363>.

@article{LianaCioban2013,

abstract = {Based on conjugate duality we construct several gap functions for general variational inequalities and equilibrium problems, in the formulation of which a so-called perturbation function is used. These functions are written with the help of the Fenchel-Moreau conjugate of the functions involved. In case we are working in the convex setting and a regularity condition is fulfilled, these functions become gap functions. The techniques used are the ones considered in [Altangerel L., Boţ R.I., Wanka G., On gap functions for equilibrium problems via Fenchel duality, Pac. J. Optim., 2006, 2(3), 667–678] and [Altangerel L., Boţ R.I., Wanka G., On the construction of gap functions for variational inequalities via conjugate duality, Asia-Pac. J. Oper. Res., 2007, 24(3), 353–371]. By particularizing the perturbation function we rediscover several gap functions from the literature. We also characterize the solutions of various variational inequalities and equilibrium problems by means of the properties of the convex subdifferential. In case no regularity condition is fulfilled, we deliver also necessary and sufficient sequential characterizations for these solutions. Several examples are illustrating the theoretical aspects.},

author = {Liana Cioban, Ernö Csetnek},

journal = {Open Mathematics},

keywords = {Variational inequalities; Gap functions; Dual gap functions; Equilibrium problems; Perturbation theory; Sequential optimality conditions; variational inequalities; gap functions; dual gap functions; equilibrium problems; perturbation theory; sequential optimality conditions},

language = {eng},

number = {5},

pages = {829-850},

title = {Revisiting the construction of gap functions for variational inequalities and equilibrium problems via conjugate duality},

url = {http://eudml.org/doc/269363},

volume = {11},

year = {2013},

}

TY - JOUR

AU - Liana Cioban

AU - Ernö Csetnek

TI - Revisiting the construction of gap functions for variational inequalities and equilibrium problems via conjugate duality

JO - Open Mathematics

PY - 2013

VL - 11

IS - 5

SP - 829

EP - 850

AB - Based on conjugate duality we construct several gap functions for general variational inequalities and equilibrium problems, in the formulation of which a so-called perturbation function is used. These functions are written with the help of the Fenchel-Moreau conjugate of the functions involved. In case we are working in the convex setting and a regularity condition is fulfilled, these functions become gap functions. The techniques used are the ones considered in [Altangerel L., Boţ R.I., Wanka G., On gap functions for equilibrium problems via Fenchel duality, Pac. J. Optim., 2006, 2(3), 667–678] and [Altangerel L., Boţ R.I., Wanka G., On the construction of gap functions for variational inequalities via conjugate duality, Asia-Pac. J. Oper. Res., 2007, 24(3), 353–371]. By particularizing the perturbation function we rediscover several gap functions from the literature. We also characterize the solutions of various variational inequalities and equilibrium problems by means of the properties of the convex subdifferential. In case no regularity condition is fulfilled, we deliver also necessary and sufficient sequential characterizations for these solutions. Several examples are illustrating the theoretical aspects.

LA - eng

KW - Variational inequalities; Gap functions; Dual gap functions; Equilibrium problems; Perturbation theory; Sequential optimality conditions; variational inequalities; gap functions; dual gap functions; equilibrium problems; perturbation theory; sequential optimality conditions

UR - http://eudml.org/doc/269363

ER -

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