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Zbl 0656.90097
Lai, Hang-Chin; Lin, Lai-Jiu
Moreau-Rockafellar type theorem for convex set functions. (English)
[J] J. Math. Anal. Appl. 132, No.2, 558-571 (1988). ISSN 0022-247X

Let (X,$\Gamma$,$\mu)$ be a finite atomless measure space and $F\sb 1,F\sb 2,...,F\sb n$, $G\sb 1,G\sb 2,...,G\sb m$ be convex real-valued set functions defined on a convex subfamily ${\cal S}$ of the $\sigma$- field $\Gamma$. Consider an optimization problem as follows: (P) Minimize $F(\Omega)=(F\sb 1(\Omega),F\sb 2(\Omega),...,F\sb n(\Omega))$ subject to $\Omega\in {\cal S}$ and $G\sb j(\Omega)\le 0$ $(j=1,2,...,m)$. The authors prove a theorem of Moreau-Rockafellar type for set functions, and then use the theorem to prove a Kuhn-Tucker type condition for an optimal solution of the minimization problem (P) for real valued set functions. If the set functions are vector-valued, the Fritz John type condition for an optimum of the multiobjective minimization problem (P) is established.
[Z.Liu]

MSC 2000:: *90C48 Programming in abstract spaces
54C60 Set-valued maps
90C25 Convex programming

Keywords: Moreau-Rockafellar theorem; finite atomless measure space; convex real- valued set functions; Kuhn-Tucker type condition; Fritz John type condition

Cited in: Zbl 0914.90232

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