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Abstract convex optimal antiderivatives. (English) Zbl 1259.47064

Starting from the previous paper [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 1, 59–66 (2011; Zbl 1251.47046)], the authors introduce and investigate \(c\)-convex \(c\)-antiderivatives and, in particular, optimal \(c\)-convex \(c\)-antiderivatives in the setting of abstract convex analysis. Their main interest is to solve the following problem: given \(f:X\to (-\infty,+\infty]\), that is, a \(c\)-antiderivative of the mapping \(M:X\rightrightarrows Y,\) and a nonempty subset \(S\) of dom\((M)\), find the \(c\)-convex solutions (\(c\)-convex extensions) \(h:X\to (-\infty,+\infty]\) of the problem \[ \text{graph}(M)\subset \text{graph}(\partial_ch):\quad h|_S=f|_S. \] The class of these functions is denoted by \(\mathcal A_{[c,f|_S,M]}\). The main definitions of \(c\)-transform, \(c\)-convexity, \(c\)-subdifferential and \(c\)-antiderivative are provided, and some basic properties of \(c\)-convex functions and \(c\)-antiderivatives, such as duality results, are recalled. In Theorem 3.2, the authors show that the class \(\mathcal A_{[c,f|_S,M]}\) is nonempty and contains both its upper envelope as well as its lower envelope. Subsequently, the notions of \(c\)-cyclic monotonicity and \(c\)-monotonicity are recalled, and the nonemptyness of \(\mathcal A_{[c,f|_S,M]}\) is reestablished by explicitly constructing, via Rockafellar’s antiderivative, the minimal \(c\)-antiderivative of \(M\) that equals \(f\) on \(S\). In the last two sections, the authors deal with Lipschitz functions on a metric space \((X,d)\) and, in particular, with the problem of the minimal and maximal Lipschitz extensions of a Lipschitz function \(f:A\subset X\to \mathbb R\) such that \[ f(y)-f(x)=d(x,y) \quad \text{for all} \;(x,y)\in \text{graph}(M), \] for a suitable map \(M:A\rightrightarrows M\). The existence of optimal extensions of the problem above is considered using abstract convexity. Notice that a proper function \(f\) is Lipschitz on \(X\) if and only if \(f=-f^c,\) where \(c=-d\) (see Proposition 5.4). In Theorem 5.7, the problem of optimal Lipschitz extensions of \(f|_S,\) which are \((-d)\)-convex \((-d)\)-antiderivatives of \(M\), is solved without requiring \(f\) to be Lipschitz outside dom\((M)\), and providing minimal and maximal solutions that are precisely the optimal antiderivatives. Finally, it is pointed out a natural minimal property of the Fitzpatrick function of a \(c\)-monotone mapping, namely, that it is a minimal antiderivative.
Reviewer: Rita Pini (Milano)

MSC:

47H04 Set-valued operators
47H05 Monotone operators and generalizations
49N15 Duality theory (optimization)
52A01 Axiomatic and generalized convexity

Citations:

Zbl 1251.47046
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References:

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