Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot equilibrium model. (English) Zbl 1191.90084
The authors make use of the Banach contraction mapping principle to prove the linear convergence of a regularization algorithm for strongly monotone Ky Fan inequalities that satisfy a Lipschitz-type condition introduced in [G. Mastroeni, in: Equilibrium problems and variational models, Nonconvex Optim. Appl. 68, 289–298 (2003; Zbl 1069.49009)]. Then, they apply the algorithm to strongly monotone Lipschitzian variational inequalities. As a consequence, they obtain a new linearly convergent derivative-free algorithm for strongly monotone complementarity problems. The linear convergence rate allows the algorithm to be coupled with inexact proximal point methods for solving monotone (not necessarily strongly monotone) problems satisfying the Lipschitz-type condition mentioned above. Finally, the authors propose a line-search free algorithm for the strong monotone problem which does not require the Lipschitz-type condition. Applications to a Nash-Cournot market equilibrium model are discussed in section 6 and some preliminary computational results are reported.
Reviewer: Regina Sandra Burachik (Adelaide)
MSC:
| 90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 49J40 | Variational inequalities |
| 49J53 | Set-valued and variational analysis |
| 65K15 | Numerical methods for variational inequalities and related problems |
| 90B50 | Management decision making, including multiple objectives |
Citations:
Zbl 1069.49009References:
| [1] | Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality III, pp. 103–113. Academic Press, New York (1972) · Zbl 0302.49019 |
| [2] | Blum, E., Oettli, W.: From optimization and variational inequality to equilibrium problems. Math. Stud. 63, 127–149 (1994) · Zbl 0888.49007 |
| [3] | Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000) · Zbl 0986.49004 |
| [4] | Mastroeni, G.: On auxiliary principle for equilibrium problems. Publicatione del Dipartimento di Mathematica dell’Universita di Pisa 3, 1244–1258 (2000) |
| [5] | Mastroeni, G.: Gap function for equilibrium problems. J. Glob. Optim. 27, 411–426 (2004) · Zbl 1061.90112 · doi:10.1023/A:1026050425030 |
| [6] | Moudafi, A.: Proximal point algorithm extended to equilibrium problem. J. Nat. Geom. 15, 91–100 (1999) · Zbl 0974.65066 |
| [7] | Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constraint equilibria, nonlinear analysis. Theory Methods Appl. 18, 1159–1166 (1992) · Zbl 0773.90092 · doi:10.1016/0362-546X(92)90159-C |
| [8] | Noor, M.A.: Auxiliary principle technique for equilibrium problems. J. Optim. Theory Appl. 122, 371–386 (2004) · Zbl 1092.49010 · doi:10.1023/B:JOTA.0000042526.24671.b2 |
| [9] | Van, N.T.T., Strodiot, J.J., Nguyen, V.H.: A bundle method for solving equilibrium problems. Math. Program. 116, 599–552 (2009) · Zbl 1155.49006 |
| [10] | Martinet, B.: Regularisation d’inéquations variationelles par approximations successives. Revue Francaise d’Automatique et d’Informatique Recherche Opérationnelle 4, 154–159 (1970) |
| [11] | Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976) · Zbl 0358.90053 · doi:10.1137/0314056 |
| [12] | Konnov, I.V.: Application of the proximal point method to nonmonotone equilibrium problems. J. Optim. Theory Appl. 119, 317–333 (2003) · Zbl 1084.49009 · doi:10.1023/B:JOTA.0000005448.12716.24 |
| [13] | Cohen, G.: Auxiliary problem principle and decomposition of optimization problems. J. Optim. Theory Appl. 32, 277–305 (1990) · Zbl 0417.49046 · doi:10.1007/BF00934554 |
| [14] | Cohen, G.: Auxiliary principle extended to variational inequalities. J. Optim. Theory Appl. 59, 325–333 (1988) · Zbl 0628.90069 · doi:10.1007/BF00940305 |
| [15] | Dinh, Q.T., Muu, L.D., Nguyen, V.H.: Extragradient methods extended to equilibrium problems. Optimization 57(6), 749–776 (2008) · Zbl 1152.90564 · doi:10.1080/02331930601122876 |
| [16] | Mangasarian, O.L., Solodov, M.V.: A linearly convergent derivative-free descant method for strongly monotone complementarity problem. Comput. Optim. Appl. 14, 5–16 (1999) · Zbl 1017.90115 · doi:10.1023/A:1008752626695 |
| [17] | Marcotte, P.: Advantages and drawbacks of variational inequalities formulations. In: Variational Inequalities and Network Equilibrium Problems, Erice, 1994, pp. 179–194. Plenum, New York (1995) · Zbl 0849.90119 |
| [18] | Murphy, H.F., Sherali, H.D., Soyster, A.L.: A mathematical programming approach for determining oligopolistic market equilibrium. Math. Program. 24, 92–106 (1982) · Zbl 0486.90015 · doi:10.1007/BF01585096 |
| [19] | Muu, L.D., Nguyen, V.H., Quy, N.V.: On the Cournot-Nash oligopolistic market equilibrium models with concave cost functions. J. Glob. Optim. 41, 351–364 (2007) · Zbl 1146.91029 · doi:10.1007/s10898-007-9243-0 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
