On gap functions and duality of variational inequality problems. (English) Zbl 0945.49004
Summary: We extend the definition of the gap function defined by Auslender for a more general class of variational inequality problems involving some convex function. A study of the duality of the extended variational inequality problem and its dual sheds new light on the meaning of gap functions. Convexity and differentiability of the gap function are also studied and sufficient conditions are derived. We also show how the gap functions for the primal and the dual are related by dual Fenchel optimization problems.
MSC:
| 49J40 | Variational inequalities |
| 90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |
Keywords:
convexity; gap function; variational inequality problems; differentiability; dual Fenchel optimization problemsReferences:
| [1] | Auslender, A., Optimisation: Méthodes Numériques (1976), Masson: Masson Paris · Zbl 0326.90057 |
| [2] | Barbu, V., Convexity and Optimization in Banach Spaces (1986), Reidel: Reidel Dordrecht · Zbl 0594.49001 |
| [3] | Fukushima, M., Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Math. Programming, 53, 99-110 (1992) · Zbl 0756.90081 |
| [4] | Hearn, D. W., The gap function of a convex program, Oper. Res. Lett., 1, 67-71 (1982) · Zbl 0486.90070 |
| [5] | Mosco, U., Dual variational inequalities, J. Math. Anal. Appl., 40, 202-206 (1972) · Zbl 0262.49003 |
| [6] | Narguney, A., Network Economics—A Variational Inequality Approach (1993), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0873.90015 |
| [7] | Rockefellar, R. T., Convex Analysis (1970), Princeton University Press · Zbl 0193.18401 |
| [8] | Stampacchia, G., Formes bilinéaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris, 258, 4413-4416 (1964) · Zbl 0124.06401 |
| [9] | Yang, X. Q., Vector variational inequality and its duality, Nonlinear Anal., 21, 869-877 (1993) · Zbl 0809.49009 |
| [10] | Zhu, D. L.; Marcotte, P., An extended descent framework for variational inequalities, J. Optim. Theory Appl., 80, 349-366 (1994) · Zbl 0798.49014 |
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