Abstract
We present an extension of Fenchel’s duality theorem by weakening the convexity assumptions to near convexity. These weak hypotheses are automatically fulfilled in the convex case. Moreover, we show by a counterexample that a further extension to closely convex functions is not possible under these hypotheses.
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Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Penot, J.P., Volle, M.: On quasiconvex duality. Math. Oper. Res. 15(4), 597–625 (1987)
Aleman, A.: On some generalizations of convex sets and convex functions. Math. Rev. Anal. Numér. Théor. Approx. 14, 1–6 (1985)
Cobzaş, Ş., Muntean, I.: Duality relations and characterizations of best approximation for p-convex sets. Rev. Anal. Numér. Théor. Approx. 16(2), 95–108 (1987)
Green, J.W., Gustin, W.: Quasiconvex sets. Can. J. Math. 2, 489–507 (1950)
Boţ, R.I., Kassay, G., Wanka, G.: Strong duality for generalized convex optimization problems. J. Optim. Theory Appl. 127(1), 45–70 (2005)
Breckner, W.W., Kassay, G.: A systematization of convexity concepts for sets and functions. J. Convex Anal. 4, 109–127 (1997)
Jeyakumar, V., Gwinner, J.: Inequality systems and optimization. J. Math. Anal. Appl. 159, 51–71 (1991)
Roberts, A.W., Varberg, D.E.: Convex Functions. Pure and Applied Mathematics, vol. 57. Academic,New York (1973)
Hamel, G.: Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: f(x+y)=f(x)+f(y). Math. Ann. 60, 459–462 (1905)
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Communicated by J.-P. Crouzeix.
The authors are grateful to the Associate Editor for helpful suggestions and remarks which improved the quality of the paper.
The second author was supported by DFG (German Research Foundation), project WA 922/1.
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Boţ, R.I., Grad, S.M. & Wanka, G. Fenchel’s Duality Theorem for Nearly Convex Functions. J Optim Theory Appl 132, 509–515 (2007). https://doi.org/10.1007/s10957-007-9234-9
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DOI: https://doi.org/10.1007/s10957-007-9234-9