Let E be a linear space, let K subset of or equal to E and f : K --> R. We formulate in terms of the lower Dini directional derivative problem 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 ( SS), the existence of a solution x* of GMVI (f', K) and the property of f to increase-along-rays starting at x*, f is an element of IAR(K, x*). We prove that the GMVI (f', K) with radially l.s.c. function f has a solution x* is an element of ker K if and only if f. IAR( K, x*). Further, we prove that the solution set of the GMVI (f', K) is a convex and radially closed subset of ker K. We show also that, if the GMVI (f', K) has a solution x* is an element of K, then x* is a global minimizer of the problem min f ( x), x is an element of 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.

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

CRESPI, GIOVANNI PAOLO;IVANOV, IVAN GINCHEV;ROCCA, MATTEO
2004

Abstract

Let E be a linear space, let K subset of or equal to E and f : K --> R. We formulate in terms of the lower Dini directional derivative problem 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 ( SS), the existence of a solution x* of GMVI (f', K) and the property of f to increase-along-rays starting at x*, f is an element of IAR(K, x*). We prove that the GMVI (f', K) with radially l.s.c. function f has a solution x* is an element of ker K if and only if f. IAR( K, x*). Further, we prove that the solution set of the GMVI (f', K) is a convex and radially closed subset of ker K. We show also that, if the GMVI (f', K) has a solution x* is an element of K, then x* is a global minimizer of the problem min f ( x), x is an element of 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.
minty variational inequalities, generalized variational inequalities, existence of solutions, increase-along-rays property, quasi-convex functions
Crespi, GIOVANNI PAOLO; Ivanov, IVAN GINCHEV; Rocca, Matteo
File in questo prodotto:
File Dimensione Formato  
fulltextjota.pdf

non disponibili

Tipologia: Documento in Post-print
Licenza: DRM non definito
Dimensione 130.17 kB
Formato Adobe PDF
130.17 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11383/6490
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 50
  • ???jsp.display-item.citation.isi??? 52
social impact