Passy, U.; Prisman, E. Z. Conjugacy in quasi-convex programming. (English) Zbl 0547.49008 Math. Program. 30, 121-146 (1984). One introduces a conjugacy relation and a subdifferential on the class of functions \(g:R^ n\to\bar R.\) It is shown that \(g^{**}=g\) iff g is proper (i.e. \(g(0)=\sup g),\) homogeneous of degree zero and evenly quasi- convex. As noted by the authors, the above notions are much related to the ones introduced by J. P. Crouzeix [Math. Oper. Res. 5, 120-125 (1980; Zbl 0428.26007)] and H. J. Greenberg and W. P. Pierskalla [J. Optimization Theory Appl. 16, 409-428 (1975; Zbl 0302.90047)], respectively, through the extension of g: \(f_ g:R^{n+1}\to\bar R\), \(f_ g(x,\lambda)=g(x/\lambda)\) if \(\lambda >0,\quad f_ g(x,\lambda)=\sup g\) if \(\lambda\leq 0.\) Properties of the conjugate and the subdifferential are stated and the relations between the conjugate and the Legendre transform. Reviewer: C.Zalinescu Cited in 21 Documents MSC: 49J45 Methods involving semicontinuity and convergence; relaxation 49N15 Duality theory (optimization) 90C30 Nonlinear programming 44A15 Special integral transforms (Legendre, Hilbert, etc.) 26B25 Convexity of real functions of several variables, generalizations Keywords:conjugacy; subdifferential; quasi-convex; Legendre transform Citations:Zbl 0428.26007; Zbl 0302.90047 PDFBibTeX XMLCite \textit{U. Passy} and \textit{E. Z. Prisman}, Math. Program. 30, 121--146 (1984; Zbl 0547.49008) Full Text: DOI References: [1] J.P. Crouzeix, ”Contributions a l’étude des functions quasiconvex”, Thesis, Université de Clermont II (France, 1977). · Zbl 0362.90096 [2] J.P. Crouzeix, ”A duality framework in quasi convex programming”, in: S. Schaible and W.T. Ziemba, eds.,Generalized concavity in optimization and economics (Academic Press, New York, 1981) pp. 207–225. [3] J.P. Crouzeix, ”Continuity and differentiability properties of quasiconvex functions on \(\mathbb{R}\)”, in: S. Schaible and W.T. Ziemba, eds.,Generalized concavity in optimization and economics (Academic Press, New York, 1981) pp. 109–130. [4] W. Fenchel, ”Convex cones, sets and functions”, Lecture Notes, Princeton University (Princeton, NJ, 1951). [5] W. Fenchel, ”A remark on convex sets and polarity”, Communication Seminar on Mathematics, University of Lung Supplementary Volume (University of Lund, Lund, 1952) 22–89. [6] J. Flachs and M.A. Pollatschek, ”Duality theorems for certain programs involving minimum or maximum operations,”Mathematical Programming 16 (1979) 348–370. · Zbl 0405.90067 [7] H.J. Greenberg and W.P. Pierskalla, ”Quasi-conjugate functions and surrogate duality”,Cahiers due Center d’Etudes de Recherche Opérationnelle 15 (1973) 437–448. · Zbl 0276.90051 [8] V.L. Klee, ”Maximal separation theorem for convex sets”,Transactions of the American Mathematical Society 134 (1968) 133–148. · Zbl 0164.52702 [9] J.E. Martinez Legaz, ”Exact quasiconvex conjugation”, Departmento de Educaciones Functionales. Universidad de Barcelona, Spain, presented at the 11th International Symposium on Mathematical Programming (Bonn, 1982). · Zbl 0522.90069 [10] W. Oettli, ”Optimality conditions involving generalized convex mappings”, in: S. Schaible and W.T. Ziemba, eds.,Generalized concavity in optimization and economics (Academic Press, New York, 1981) pp. 227–238. · Zbl 0538.90080 [11] E.Z. Prisman, ”A new approach to duality Lagrangians and saddle functions in quasi convex programming”, Ph.D. Dissertation, Technion (Haifa, Israel, 1982) (in Hebrew). [12] R.T. Rockafellar,Convex analysis (Princeton University Press, Princeton, NJ, 1970). · Zbl 0193.18401 [13] I. Singer, ”Pseudo-conjugate functionals and pseudo-duality”, in: Proceedings of the International Conference in Mathematical Methods in Operations Research Sofia (Publishing House of the Bulgarian Academy of Science, Sofia, 1980). · Zbl 0478.90079 [14] Y.I. Zabotin, A.I. Korablev and R.F. Khabibullin, ”Conditions for an extremum of a functional in case of constraints”,Cybernetics 9 (1975) 982–988. 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.