Minty variational inequalities, increase-along-rays property and optimization.

*(English)*Zbl 1059.49010Summary: Let \(E\) be a linear space, let \(K\subseteq E\) and \(f:K \to\mathbb{R}\). We formulate in terms of the lower Dini directional derivative the problem of the generalized Minty variational inequality GMVI \((f',K)\), which can be considered as a generalization of MVI\((f',K)\), the Minty variational inequality of differential type. We investigate, in the case of \(K\) star-shaped, the existence of a solution \(x^*\) of GMVI\((f',K)\) and the property of \(f\) to increase-along-rays starting at \(x^*\) \(f\in\text{IAR}(K,x^*)\). We prove that the GMVI\((f',K)\) with radially l.s.c. function \(f\) has a solution \(x^*\in\text{ker}\,K\) if and only if \(f\in\text{IAR}(K,x^*)\). Further, we prove that the solution set of the GMVI \((f',K)\) is a convex and radially closed subset of \(\text{ker}\,K\). We show also that, if the GMVT\((f',K)\) has a solution \(x^*\in K\), then \(x^*\) is a global minimizer of the problem \(\min f(x)\), \(x\in K\). Moreover, we observe that the set of the global minimizers of the related optimization problem, its kernel, and the solution set of the variational inequality can be different. Finally, we prove that, in the case of a quasiconvex function \(f\), these sets coincide.

##### MSC:

49J40 | Variational inequalities |

47J20 | Variational and other types of inequalities involving nonlinear operators (general) |

47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |

##### Keywords:

Minty variational inequalities; generalized variational inequalities; existence of solutions; increase-along-rays property; quasi-convex functions
PDF
BibTeX
XML
Cite

\textit{G. P. Crespi} et al., J. Optim. Theory Appl. 123, No. 3, 479--496 (2004; Zbl 1059.49010)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Baiocchi, C., and Capelo, A., Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, John Wiley and Sons, New York, NY, 1984. · Zbl 0551.49007 |

[2] | Kinderlehrer, D., and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, NY, 1980. · Zbl 0457.35001 |

[3] | Giannessi, F., Theorems of the Alternative, Quadratic Programs, and Complementarity Problems, Variational Inequality and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, John Wiley and Sons, New York, NY, pp. 151-186, 1980. · Zbl 0484.90081 |

[4] | Giannessi, F., On Minty Variational Principle, New Trends in Mathematical Programming, Edited by F. Giannessi, S. Komlósi, and T. Rapcsak, Kluwer, Dordrecht, Holland, pp. 93-99, 1998. · Zbl 0909.90253 |

[5] | Komlósi, S., On the Stampacchia and Minty Variational Inequalities, Generalized Convexity and Optimization for Economic and Financial Decisions, Edited by G. Giorgi and F. Rossi (Proceedings of the Workshop held in Verona, Italy, 28-29 May 1998), Pitagora Editrice, Bologna, Italy, pp. 231-260, 1999. |

[6] | Lions, J. L., and Stampacchia, G., Variational Inequalities, Communications on Pure and Applied Mathematics, Vol. 20, pp. 493-512, 1967. · Zbl 0152.34601 |

[7] | Minty, G. J., On the Generalization of a Direct Method of the Calculus of Variations, Bulletin of the American Mathematical Society, Vol. 73, pp. 314-321, 1967. · Zbl 0157.19103 |

[8] | Stampacchia, G., Formes Bilin’eaires Coercives sur les Ensembles Convexes, Comptes Rendus de l’Acad’emie des Sciences de Paris, Groupe 1, Vol. 258, pp. 4413-4416, 1964. · Zbl 0124.06401 |

[9] | Rubinov, A. M., Abstract Convexity and Global Optimization, Kluwer, Dordrecht, Holland, 2000. · Zbl 0985.90074 |

[10] | Mastroeni, G., Some Remarks on the Role of Generalized Convexity in the Theory of Variational Inequalities, Generalized Convexity and Optimization for Economic and Financial Decisions, Edited by G. Giorgi and F. Rossi (Proceedings of the Workshop held in Verona, Italy, 28-29 May 1998), Pitagora Editrice, Bologna, Italy. pp. 271-281, 1999. |

[11] | Mordukhovich, B., Stability Theory for Parametric Generalized Equations and Variational Inequalities via Nonsmooth Analysis, Transactions of the American Mathematical Society, Vol. 343, pp. 609-657, 1994. · Zbl 0826.49008 |

[12] | Thach, P. T., and Kojima, M., A Generalized Convexity and Variational Inequality for Quasiconvex Minimization, SIAM Journal on Optimization, Vol. 6, pp. 212-226, 1996. · Zbl 0841.49019 |

[13] | Yang, X. Q., Generalized Convex Functions and Vector Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 79, pp. 563-580, 1993. · Zbl 0797.90085 |

[14] | Crespi, G. P., Ginchev, I., and Rocca, M., Existence of Solutions and Star-Shapedness in Minty Variational Inequalities, Journal of Global Optimization (to appear). · Zbl 1097.49007 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.