## $$*$$-dualities.(English)Zbl 0815.49026

Summary: For a complete, totally ordered group $$A= (A,\leq, *)$$ [in the sense of G. Birkhoff: “Lattice theory” (1948; Zbl 0033.101), Chapter 14], let $$\overline A= (\overline A, \leq,\overline {*},\underline {*})$$, where $$\overline A= A\cup \{+\infty\}\cup \{-\infty\}$$, with $$\leq$$ and $$*$$ extended from $$A$$ to $$\leq$$ and, respectively, $$\overline {*}$$ and $$\underline {*}$$, on $$\overline A$$, in a natural way, and, for any set $$X$$, extend $$\leq$$, $$\overline {*}$$ and $$\underline {*}$$ from $$\overline A$$ to $$\overline A^ X$$, pointwise on $$X$$. For two sets $$X$$ and $$W$$, we introduce and study $$*$$-dualities $$M: f\in \overline A^ X\to f^ M\in \overline A^ W$$ (by the conditions $$(\inf_{i\in I} f_ i)^ M= \sup_{i\in I} f^ M_ i$$ and $$(f\overline {*} a)^ M= f^ M{\underline {*}}a^{-1}$$ for all $$\{f_ i\}_{i\in I}\subseteq \overline A^ X$$, $$f\in \overline A^ X$$, $$a\in A$$), which encompass various kinds of known conjugations and polarities. We give some applications to $$(A,\leq,*)= (R, \leq, +)$$, $$(A, \leq,*)= (R_ +\backslash\{0\}, \leq, \times)$$, and to duality in optimization.

### MSC:

 49N15 Duality theory (optimization) 06A15 Galois correspondences, closure operators (in relation to ordered sets) 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 90C26 Nonconvex programming, global optimization

Zbl 0033.101
Full Text:

### References:

  Aggeri J.C., Comptes Rendus Acad.Sci.Paris 260 pp 6011– (1965)  Aggeri J. C., Sé; min. Math. Fac. Sci. Montpellier (1965)  Birkhoff G., Amer.Math.Soc.Coll.Publ 25 (1948)  DOI: 10.1080/02331939208843805 · Zbl 0817.90122  Elster K.H., Lecture Notes in Econ. and Math Systems 345 pp 219– (1990)  DOI: 10.1007/BF02589354 · Zbl 0064.10504  Martinez-Legaz J.E., Lecture Notes in Econ. and Math. Systems 345 pp 168– (1990)  DOI: 10.1080/02331939008843573 · Zbl 0728.90071  DOI: 10.1007/978-3-0348-8625-3_30  Moreau J J., Sémin Eq Dériv Part 2 (1966)  Moreau J J., J Math Pures Appl 49 pp 109– (1970)  Rockafellar R T., Convex analysis (1970) · Zbl 0193.18401  DOI: 10.1017/S0004972700010844 · Zbl 0404.90101  Singer I., Lecture Note in Econ and Math Systems 226 pp 80– (1984)  Singer I., Lecture notes in Pure and Appl Math pp 253–  Tind J., Math Progr Study 12 pp 206– (1980) · Zbl 0437.90095  DOI: 10.1016/0022-247X(78)90243-3 · Zbl 0403.90066  Volle M., Contributions à la dualité en optimisation et à lépi-convergence (1986)  Dolecki, S. Remarks on duality theory In Abstracts internat. confer on mathematical programming-theory and applications. pp.28–31. Wartburg Eisenach  Evers J.J.M., Nieuw Arch Wisk 3 pp 23– (1985)  DOI: 10.1016/0022-247X(81)90178-5 · Zbl 0479.46003  DOI: 10.1080/02331938208842813 · Zbl 0529.90096
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.