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Duality principles for optimization problems dealing with the difference of vector-valued convex mappings. (English) Zbl 1041.90068
Consider the following equivalent problems: $\min f(x)+g\bigl( G(x)-H(x) \bigr), \text{ on a vectorial space},\tag{P}$ and $\min f(x), \text{ for } G(x)-H(x)\leq 0,\tag{R}$ where $$f,g$$ are convex functions and $$G,H$$ are vector-valued mappings that are convex with respect to a partial vectorial order for which $$g$$ is nondecreasing.
In this paper the author obtains duality formulas for problems (P) and (R) in terms of the Legendre-Fenchel conjugates of the data functions. The author also provides relations between the optimal solutions of primal and dual problems and a general necessary optimality condition. In particular the author applies the results to the problem of minimization of the composite of a convex mapping with a nonincreasing convex function and the minimization of the upper envelope of a family of concave functions.

##### MSC:
 90C46 Optimality conditions and duality in mathematical programming 49J52 Nonsmooth analysis 49N15 Duality theory (optimization)
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##### References:
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