New properties of the Variational Sum of monotone operators

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We study the Variational Sum of monotone operators, in particular its relationship with the Extended Sum of monotone operators. First, we establish some new properties of the Variational Sum, among them that this sum has closed graph and convex values. Then, we show that the graph of the Variational Sum always contains the graph of the Extended Sum, and hence, it contains also the graph of the usual sum. An example is given showing that the latter inclusions are proper in general.
Original languageEnglish
Pages (from-to)767-778
Number of pages12
JournalJournal of Convex Analysis
Issue number3-4
StatePublished - 1 Dec 2009
Externally publishedYes


  • Extended sum
  • Monotone operator
  • Variational Sum


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