The concept of cyclic monotonicity of a multivalued map has been introduced by R. T. Rockafellar with reference to the subdifferential operator of a convex functional. He observed that cyclic monotonicity could be viewed heuristically as a discrete substitute for the classical condition of conservativity, i.e., the vanishing of all the circuital integrals of a vector field. In the present paper a potential theory for monotone multivalued operators is developed, and, in this context, an answer to Rockafellar's conjecture is provided. It is first proved that the integral of a monotone multivalued map along lines and polylines can be properly defined. This result allows us to introduce the concept of conservativity of.a monotone multivalued map and to state its relation with cyclic monotonicity. Further, as a generalization of a classical result of integral calculus, it is proved that the potential of the subdifferential of a convex functional coincides, to within an additive constant, with the restriction of the functional on the domain of its subdifferential map. It is then shown that any conservative monotone graph admits a pair of proper convex potentials which meet a complementarity relation. Finally, sufficient conditions are given under which the complementary and the Fenchel's conjugate of the potential associated with a conservative maximal monotone graph do coincide.

Romano, G., Rosati, L., Sciarra, F., Bisegna, P. (1993). A potential theory for monotone multivalued operators. QUARTERLY OF APPLIED MATHEMATICS, 51(4), 613-631.

A potential theory for monotone multivalued operators

ROSATI, LUCA;BISEGNA, PAOLO
1993-01-01

Abstract

The concept of cyclic monotonicity of a multivalued map has been introduced by R. T. Rockafellar with reference to the subdifferential operator of a convex functional. He observed that cyclic monotonicity could be viewed heuristically as a discrete substitute for the classical condition of conservativity, i.e., the vanishing of all the circuital integrals of a vector field. In the present paper a potential theory for monotone multivalued operators is developed, and, in this context, an answer to Rockafellar's conjecture is provided. It is first proved that the integral of a monotone multivalued map along lines and polylines can be properly defined. This result allows us to introduce the concept of conservativity of.a monotone multivalued map and to state its relation with cyclic monotonicity. Further, as a generalization of a classical result of integral calculus, it is proved that the potential of the subdifferential of a convex functional coincides, to within an additive constant, with the restriction of the functional on the domain of its subdifferential map. It is then shown that any conservative monotone graph admits a pair of proper convex potentials which meet a complementarity relation. Finally, sufficient conditions are given under which the complementary and the Fenchel's conjugate of the potential associated with a conservative maximal monotone graph do coincide.
1993
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore ICAR/08 - SCIENZA DELLE COSTRUZIONI
Settore MAT/05 - ANALISI MATEMATICA
English
Con Impact Factor ISI
convex analysis; multivalued operators; ciclically monotone operators
http://www.ams.org/mathscinet/
Romano, G., Rosati, L., Sciarra, F., Bisegna, P. (1993). A potential theory for monotone multivalued operators. QUARTERLY OF APPLIED MATHEMATICS, 51(4), 613-631.
Romano, G; Rosati, L; Sciarra, F; Bisegna, P
Articolo su rivista
File in questo prodotto:
File Dimensione Formato  
MonotoneOperators_QAM.pdf

solo utenti autorizzati

Descrizione: Articolo
Dimensione 731.91 kB
Formato Adobe PDF
731.91 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: https://hdl.handle.net/2108/13597
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 22
  • ???jsp.display-item.citation.isi??? 20
social impact