## $$\bigvee$$-dualities and $$\perp$$-dualities.(English)Zbl 0728.06007

Summary: We introduce and study dualities $$\Delta$$ : $${\bar {\mathbb{R}}}^ X\to {\bar {\mathbb{R}}}^ W$$ (i.e., mappings $$f\in {\bar {\mathbb{R}}}^ X\to f^{\Delta}\in {\bar {\mathbb{R}}}^ W$$ such that $$(\inf_{i\in I}f_ i)^{\Delta}=\sup_{i\in I}f_ i^{\Delta}$$ for all $$\{f_ i\}_{i\in I}\subseteq {\bar {\mathbb{R}}}^ X$$ and all index sets I), which satisfy the additional condition $$(f\vee d)^{\Delta}=f^{\Delta}\wedge -d$$ (f$$\in {\bar {\mathbb{R}}}^ X$$, $$d\in {\bar {\mathbb{R}}})$$, and their duals, which are characterized as those dualities $$\Delta^*: {\bar {\mathbb{R}}}^ W\to {\bar {\mathbb{R}}}^ X$$ for which $$(f\perp d)^{\Delta^*}=f^{\Delta^*}\top -d$$ (f$$\in {\bar {\mathbb{R}}}^ X$$, $$d\in {\bar {\mathbb{R}}})$$, where $$\perp$$ and $$\top$$ are two new binary operations on $${\bar {\mathbb{R}}}$$, which we introduce here. Furthermore, we give a characterization of those $$\Delta$$ which are also conjugations. Some applications are also mentioned.

### MSC:

 06A15 Galois correspondences, closure operators (in relation to ordered sets) 06B23 Complete lattices, completions

### Keywords:

dualities; conjugations
Full Text:

### References:

 [1] Birkhoff G., Lattice theory 25 (1948) [2] DOI: 10.1137/0316018 · Zbl 0397.46013 [3] Elster K.-H, Optimal control Lecture Notes in Control and Information Sci. 95 pp 79– (1987) [4] Elster K.H., On a general concept of conjugate functions as an approach to non-convex optimization problem (1987) [5] Elster K.H., Trends in mathematical optimization. ISNM 84 pp 67– (1988) · Zbl 0655.90080 [6] Evers J.J.M., Nieuw Arch. Wisk 3 pp 23– (1985) [7] DOI: 10.1007/BF01582120 · Zbl 0405.90067 [8] Greenberg H.J., Cahiers Centre d’Et Rech. Opér 15 pp 437– (1973) [9] DOI: 10.1080/02331938808843379 · Zbl 0671.49015 [10] Martinez-Legaz, J.E. Generalized conjugation and related topics. Proc. Internal. Workshop on Generalized concavity, fractional programming and economic applications. 30 May1988, Pisa. Jun 1 (to appear) · Zbl 0861.49029 [11] DOI: 10.1080/02331939008843573 · Zbl 0728.90071 [12] Martinez-Legaz J.-E, Some characterizations of surrogate dual problems · Zbl 0817.90112 [13] Moreau J.J., Fonctionnelles convexes 2 (1966) [14] Moreau J.J., J. Math. Pures Appl 49 pp 109– (1970) [15] Ore O., Theory of graphs 38 (1962) · Zbl 0105.35401 [16] DOI: 10.1007/BF01899228 · Zbl 0047.26402 [17] DOI: 10.1007/BF00938758 · Zbl 0542.90083 [18] Singer, I. Pseudo-conjugate functionals and pseudo-duality. Mathematical methods in operations research (invited lectures presented at the Internat. Confer, in Sofia. Nov1980. pp.115–134. Sofia: Publ. House Bulgarian Acad. Sci. [19] DOI: 10.1016/0362-546X(83)90020-2 · Zbl 0528.49007 [20] DOI: 10.1080/00036818308839476 · Zbl 0526.90097 [21] Singer I., Selected topics in operations research and mathematical economics 226 pp 80– (1984) [22] DOI: 10.1016/0362-546X(84)90033-6 · Zbl 0538.49005 [23] DOI: 10.1016/0022-247X(86)90021-1 · Zbl 0601.46043 [24] DOI: 10.1007/BF01774294 · Zbl 0638.06006 [25] Volle M., Contributions à la dualite en optimisation et a Fepi-convergcnce (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.