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Conjugate functionals and level sets. (English) Zbl 0538.49005

Let E be a locally convex space with conjugate space \(E^*\). For any \(f:E\to \bar R=[-\infty +\infty]\) and \(\lambda\in R\), the \(\lambda\)- quasi-conjugate of f in the sense of H. J. Greenberg and W. P. Pierskalla [Cahiers Centre Etud. Rech. Opér. 15, 437-448 (1973; Zbl 0276.90051)] is the functional \(f^{\gamma}_{\lambda}:E^*\to \bar R\) defined by \(f^{\gamma}_{\lambda}(\Phi)=\lambda -\inf \{f(x)| x\in E,\Phi(x)\geq \lambda \} (\Phi \in E^*)\) and the second \(\lambda\)-quasi- conjugate \((f^{\gamma}_{\lambda})^{\gamma}_{\lambda}:E\to \bar R\) is defined similarly; also, by definition, \(f^{\gamma \gamma}=\sup_{\lambda \in R}(f^{\gamma}_{\lambda})^{\gamma}_{\lambda}.\) These conjugates are useful in quasi-convex optimization. J. P. Crouzeix [”Contributions à l’étude des fonctions quasi-convexes”, Thesis (Clermont 1977)] has observed that these notions do not make use of the level sets of f, important in the study of quasi-convexity, and therefore he has introduced analogous notions, involving level sets.
In the present paper we give some new expressions of the \(\lambda\)-quasi- conjugates of f, the \(\lambda\)-pseudo-conjugates of f [the author, Math. Meth. Oper. Res., int. Conf. Sofia 1980, 115-134 (1981; Zbl 0478.90079)], the \(\lambda\)-semi-conjugates of f [the author, Nonlinear Anal., Theory Methods Appl. 7, 1115-1121 (1983; Zbl 0528.49007)], and the usual Fenchel conjugate of f, with the aid of the level sets \(A_ c=\{x\in E| f(x)<c\},\quad S_ c=\{x\in E| f(x)\leq c\}\quad(c\in R)\) and corresponding results for the higher conjugates of f. For example (part of theorem 2), \[ f^{\gamma}_{\lambda}(\Phi)=\inf \{\nu \in R| \Phi(x)<\lambda \quad(x\in A_{\lambda -\nu})\}=\inf \{\nu \in R| \Phi(x)<\lambda \quad(x\in S_{\lambda -\nu})\}\quad(\Phi \in E^*,\lambda \in R). \] Also, we determine the level sets of these conjugates. For example (part of theorem 4), \[ S_ c(f^{\gamma}_{\lambda})=\cap_{x\in A_{\lambda -c}}\{\Phi \in E^*| \Phi(x)<\lambda \}\quad(c,\lambda \in R). \]

MSC:

49N15 Duality theory (optimization)
90C25 Convex programming
46A03 General theory of locally convex spaces
49J45 Methods involving semicontinuity and convergence; relaxation
90C48 Programming in abstract spaces
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[1] Crouzeix, J.-P., Contributions à l’étude des fonctions quasi-convexes, (thèse (1977), Université de Clermont)
[2] Crouzeix, J.-P., Conjugacy in quasiconvex analysis, (Convex Analysis and its Applications. Convex Analysis and its Applications, Lecture Notes in Economics and Mathematical Systems, 144 (1977), Springer: Springer Berlin), 66-99 · Zbl 0362.90096
[3] Crouzeix, J.-P., A duality framework in quasi-convex programming, (General Concavity in Optimization and Economics (1981), Academic Press: Academic Press New York), 207-225
[4] Greenberg, H. J.; Pierskalla, W. P., Quasi-conjugate functions and surrogate duality, Cah. Cent. Rech. Opér., 15, 437-448 (1973) · Zbl 0276.90051
[5] Klee, V., Maximal separation theorems for convex sets, Trans. Am. math. Soc., 134, 133-147 (1968) · Zbl 0164.52702
[6] Rockafellar, R. T., Level sets and continuity of conjugate convex functions, Trans. Am. math. Soc., 123, 46-63 (1966) · Zbl 0145.15802
[7] Singer, I., Pseudo-conjugate functionals and pseudo-duality, (Mathematical Methods in Operations Research (1981), Publ. House Bulgarian Acad. Sci: Publ. House Bulgarian Acad. Sci Sofia), 115-134
[8] Singer, I., Optimization by level set methods. IV: Generalizations and complements, Numer. funct. Analysis Optim., 4, 279-310 (1981-1982) · Zbl 0497.49022
[9] Singer, I., The lower semi-continuous quasi-convex hull as a normalized second conjugate, Nonlinear Analysis, 7, 1115-1121 (1983) · Zbl 0528.49007
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