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

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