We introduce a new representation for maximal monotone operators. We relate it to previous representations given by Krauss, Fitzpatrick and Martı́nez-Legaz and Théra. We show its usefulness for the study of compositions and sums of maximal monotone operators.
Nous introduisons une représentation nouvelle pour les opérateurs maximaux monotones à l'aide de fonctions convexes. Nous la relions à des représentations dues à Krauss, Fitzpatrick, Martı́nez-Legaz et Théra. Nous montrons son utilité pour obtenir des règles de composition et de somme.
Accepted:
Published online:
Jean-Paul Penot 1
@article{CRMATH_2004__338_11_853_0, author = {Jean-Paul Penot}, title = {A representation of maximal monotone operators by closed convex functions and its impact on calculus rules}, journal = {Comptes Rendus. Math\'ematique}, pages = {853--858}, publisher = {Elsevier}, volume = {338}, number = {11}, year = {2004}, doi = {10.1016/j.crma.2004.03.017}, language = {en}, }
TY - JOUR AU - Jean-Paul Penot TI - A representation of maximal monotone operators by closed convex functions and its impact on calculus rules JO - Comptes Rendus. Mathématique PY - 2004 SP - 853 EP - 858 VL - 338 IS - 11 PB - Elsevier DO - 10.1016/j.crma.2004.03.017 LA - en ID - CRMATH_2004__338_11_853_0 ER -
Jean-Paul Penot. A representation of maximal monotone operators by closed convex functions and its impact on calculus rules. Comptes Rendus. Mathématique, Volume 338 (2004) no. 11, pp. 853-858. doi : 10.1016/j.crma.2004.03.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.03.017/
[1] On the maximality of the sum of two maximal monotone operators, Nonlinear Anal., Volume 5 (1981) no. 2, pp. 143-147
[2] Duality for the sum of convex functions in general Banach spaces (J.A. Barroso, ed.), Aspects of Mathematics and its Applications, North-Holland, Amsterdam, 1986, pp. 125-133
[3] Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1971
[4] ε-enlargements of maximal monotone operators in Banach spaces, Set-Valued Anal., Volume 7 (1999) no. 2, pp. 117-132
[5] Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal., Volume 10 (2002) no. 4, pp. 297-316
[6] Representing monotone operators by convex functions, Functional Analysis and Optimization, Workshop and Miniconference, Canberra, Australia, 1988, Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 20, 1988, pp. 59-65
[7] A representation of maximal monotone operators by saddle functions, Rev. Roumainl Math. Pures Appl., Volume 30 (1985), pp. 823-836
[8] A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions, Nonlinear Anal., Volume 9 (1985), pp. 1381-1399
[9] Maximal monotone operators and saddle functions. I, Z. Anal. Anwend., Volume 5 (1986), pp. 333-346
[10] A convex representation of maximal monotone operators, J. Nonlinear and Convex Anal., Volume 2 (2001) no. 2, pp. 243-247
[11] Minimax and Monotonicity, Lecture Notes in Math., vol. 1693, Springer, Berlin, 1998
[12] Sum theorems for monotone operators and convex functions, Trans. Amer. Math. Soc., Volume 350 (1998) no. 7, pp. 2953-2972
[13] A family of enlargements of maximal monotone operators, Set-Valued Anal., Volume 8 (2000) no. 4, pp. 311-328
[14] Convex Analysis in General Vector Spaces, World Scientific, Singapore, 2002
Cited by Sources:
Comments - Policy