Characterization of Nonsmooth Semistrictly Quasiconvex and Strictly Quasiconvex Functions

New concepts of semistrict quasimonotonicity and strict quasimonotonicity for multivalued maps are introduced. It is shown that a locally Lipschitz map is (semi)strictly quasiconvex if and only if its Clarke subdifferential is (semi)strictly quasimonotone. Finally, an existence result for the corresponding variational inequality problem is obtained.

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