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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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