Journal of Convex Analysis 33 (2026), No. 3&4, 903--924
Copyright Heldermann Verlag 2026
Characterizations of Variational Convexity and Tilt Stability via Quadratic Bundles
Pham Duy Khanh
Dept. of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam
khanhpd@hcmue.edu.vn
Boris S. Mordukhovich
Dept. of Mathematics, Wayne State University, Detroit, Michigan, U.S.A.
aa1086@wayne.ed
Vo Thanh Phat
Dept. of Mathematics & Statistics, University of North Dakota, Grand Forks, North Dakota, U.S.A.
thanh.vo.1@und.edu
Le Duc Viet
Dept. of Mathematics, Wayne State University, Detroit, Michigan, U.S.A.
vietle@wayne.edu
We establish characterizations of variational s-convexity and tilt stability for prox-regular functions in the absence of subdifferential continuity via quadratic bundles, a kind of primal-dual generalized second-order derivatives recently introduced by Rockafellar. Deriving such characterizations in the effective pointbased form requires a certain revision of quadratic bundles investigated below. Our device is based on the notion of generalized twice differentiability and its novel characterization via classical twice differentiability of the associated Moreau envelopes combined with various limiting procedures for functions and sets.
Keywords: Set-valued and variational analysis, prox-regularity, variational convexity, tilt stability, epi-convergence, subderivatives, generalized twice differentiability, quadratic bundles, Moreau envelopes.
MSC: 49J52, 49J53, 90C31.
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