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On certain optimization problems in infinite-dimensional spaces. (English) Zbl 1097.49022

Let \((X,\mathcal A,\mu)\) be an atomless finite positive measure space. A set \(A\in \mathcal A\) can be identified with its characteristic function \(I_A\in L_\infty(X,\mathcal A,\mu).\)
R. J. T. Morris [J. Math. Anal. Appl. 70, 546–562 (1979; Zbl 0417.49032)] has defined a notion of convexity for subsets of \(\mathcal A\) (or, equivalently, of \( L_\infty(X,\mathcal A,\mu)\)) and for real-valued functions defined on such sets, and applied it to study constrained optimization problems for functions defined on families of measurable sets.
The authors of the present paper show that this notion can be transposed to subsets of locally convex spaces, being closely related to that of sets with convex closure. They show that a Morris convex function is upper semi-continuous, develop a convex analysis (conjugation, subdifferentials, etc.) for these kind of functions, and apply these results to constrained optimization problems with Morris convex functions defined on sets with convex closure. Morris quasi-convex and Morris pseudo-convex functions are also considered.

MSC:

49K27 Optimality conditions for problems in abstract spaces
49J53 Set-valued and variational analysis
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Citations:

Zbl 0417.49032
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