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M. Glover, “Duality for increasing positively homogeneous functions and normal sets,” Recherche Op´erationnelle/Oper. , 32, 105–123 (1998). 16. A. M. Rubinov and B. S¸im¸sek, “Dual problems of quasiconvex maximization,” Bull. Aust. Math. , 51, 139–144 (1995). 17. A. M. Rubinov and B. S ¸ im¸sek, “Conjugate quasiconvex nonnegative functions,” Optimization, 35, 1–22 (1995). 18. I. Singer, “Conjugation operators,” In: Selected Topics in Operations Research and Mathematical Economics, Lect. Notes Econ.

2) In this section, we consider the particular case where ∆ is a (∗, s)-duality with ∗ : A × A → A and s : A → A as in the preceding sections. 1. Let X and W be two sets, A = (A, ) be a complete lattice endowed with a binary s operation ∗ : A × A → A satisfying condition (inf-d), s : A → A be an antiautomorphism such that (∗−1 u ) X W admits a left neutral element, and ∆ : A → A be a (∗, s)-duality. 3) and for any such ϕ and any f ∈ AX and w ∈ W , we have ∆ ∂ ∆ f (x0 ) = {w ∈ W | ϕ (w, x0 ) ∗−1 u f (w) = f (x0 )}.

Kopytov, Fully Ordered Groups [in Russian], Nauka, Moscow (1972); English translation: Wiley (1974). 7. -E. Mart´ınez-Legaz and I. Singer, “Dualities between complete lattices,” Optimization, 21, 481–508 (1990). 8. -E. Mart´ınez-Legaz and I. Singer, “∨-Dualities and ⊥-dualities,” Optimization, 22, 483–511 (1991). 9. -E. Mart´ınez-Legaz and I. Singer, “∗-Dualities,” Optimization, 30, 295–315 (1994). 10. -E. Mart´ınez-Legaz and I. Singer, “Dualities associated to binary operations on R,” J. , 2, 185–209 (1995).

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