## Moreau-Rockafellar type theorem for convex set functions.(English)Zbl 0656.90097

Let (X,$$\Gamma$$,$$\mu)$$ be a finite atomless measure space and $$F_ 1,F_ 2,...,F_ n$$, $$G_ 1,G_ 2,...,G_ m$$ be convex real-valued set functions defined on a convex subfamily $${\mathcal S}$$ of the $$\sigma$$- field $$\Gamma$$. Consider an optimization problem as follows: (P) Minimize $$F(\Omega)=(F_ 1(\Omega),F_ 2(\Omega),...,F_ n(\Omega))$$ subject to $$\Omega\in {\mathcal S}$$ and $$G_ j(\Omega)\leq 0$$ $$(j=1,2,...,m)$$. 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.
Reviewer: Z.Liu

### MSC:

 90C48 Programming in abstract spaces 54C60 Set-valued maps in general topology 90C25 Convex programming
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### References:

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