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Subdifferential characterization of quasiconvexity and convexity. (English) Zbl 0832.49010
Summary: Let \(f : X \to \mathbb{R} \cup \{+ \infty\}\) be a lower semicontinuous function on a Banach space \(X\). We show that \(f\) is quasiconvex if and only if its Clarke subdifferential \(\partial f\) is quasimonotone. As an immediate consequence, we get that \(f\) is convex if and only if \(\partial f\) is monotone.

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
49J52 Nonsmooth analysis
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