Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Set-valued and variational analysis
  • prox-regularity
  • variational convexity
  • tilt stability
  • epi-convergence
  • subderivatives
  • generalized twice differentiability
  • quadratic bundles
  • Moreau envelopes

Mathematics Subject Classification

49J52, 49J53, 90C31

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Suggested citation

P. D. Khanh, B. S. Mordukhovich, V. T. Phat, L. D. Viet. “Characterizations of Variational Convexity and Tilt Stability via Quadratic Bundles.” Journal of Convex Analysis 33 (2026), No. 3&4, 903–924. https://doi.org/10.68381/jca33050

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Published by Heldermann Verlag, 2026. Rights now held by Banach Press LLC.