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On the use of semi-closed sets and functions in convex analysis. (English) Zbl 1308.90132

Summary: The main aim of this short note is to show that the subdifferentiability and duality results established by M. Laghdir [Appl. Math. E-Notes 5, 150–156 (2005; Zbl 1085.49022)], M. Laghdir and R. Benabbou [Appl. Math. Sci., Ruse 1, No. 21–24, 1019–1033 (2007; Zbl 1200.90138)], and M. Alimohammady et al. [Appl. Math. Lett. 24, No. 8, 1289–1294 (2011; Zbl 1219.26011)], stated in Fréchet spaces, are consequences of the corresponding known results using Moreau-Rockafellar type conditions.

MSC:

90C25 Convex programming
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References:

[1] Alimohammady, M., Cho, Y.J., Dadashi, V., Roohi, M., Convex sub-differential sum rule via convex semi-closed functions with applications in convex programming, Appl. Math. Lett., 2011, 24(8), 1289-1294; · Zbl 1219.26011
[2] Laghdir, M., Some remarks on subdifferentiability of convex functions, Appl. Math. E-Notes, 2005, 5, 150-156 (electronic); · Zbl 1085.49022
[3] Laghdir, M., Benabbou, R., Convex functions whose epigraphs are semi-closed: duality theory, Appl. Math. Sci. (Ruse), 2007, 1(21-24), 1019-1033; · Zbl 1200.90138
[4] Zălinescu, C., Convex Analysis in General Vector Spaces, World Scientific, River Edge, 2002; · Zbl 1023.46003
[5] Zălinescu, C., Hahn-Banach extension theorems for multifunctions revisited, Math. Methods Oper. Res., 2008, 68(3), 493-508; · Zbl 1171.90016
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