Publications

2025

1. preprint

   Characterizations of Variational Convexity and Tilt Stability via Quadratic Bundles

   P. D. Khanh, B. S. Mordukhovich, V. T. Phat, and L. D. Viet

   arXiv preprint, 2025

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   In this paper, we establish characterizations of variational -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.

   @article{quadbund-char,
     title = {Characterizations of Variational Convexity and Tilt Stability via Quadratic Bundles},
     author = {Khanh, P. D. and Mordukhovich, B. S. and Phat, V. T. and Viet, L. D.},
     journal = {arXiv preprint},
     year = {2025},
     url = {https://arxiv.org/abs/2501.04629},
     dimensions = {true},
   }

2. preprint

   Generalized Twice Differentiability and Quadratic Bundles in Second-Order Variational Analysis

   P. D. Khanh, B. S. Mordukhovich, V. T. Phat, and L. D. Viet

   arXiv preprint, 2025

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   In this paper, we investigate the concepts of generalized twice differentiability and quadratic bundles of nonsmooth functions that have been very recently proposed by Rockafellar in the framework of second-order variational analysis. These constructions, in contrast to second-order subdifferentials, are defined in primal spaces. We develop new techniques to study generalized twice differentiability for a broad class of prox-regular functions, establish their novel characterizations. Subsequently, quadratic bundles of prox-regular functions are shown to be nonempty, which provides the ground of potential applications in variational analysis and optimization.

   @article{quadbund,
     title = {Generalized Twice Differentiability and Quadratic Bundles in Second-Order Variational Analysis},
     author = {Khanh, P. D. and Mordukhovich, B. S. and Phat, V. T. and Viet, L. D.},
     journal = {arXiv preprint},
     year = {2025},
     url = {https://arxiv.org/abs/2501.02067},
     dimensions = {true},
   }

3. journal

   Lipschitz Modulus of Convex Functions via Function Values

   P. D. Khanh, V. V. H. Khoa, V. T. Phat, and L. D. Viet

   Optimization Letters, 2025

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   In this note, we establish the Lipschitz continuity of finite-dimensional globally convex functions on all given balls and global Lipschitz continuity for eligible functions of that type. The Lipschitz constants in both situations draw information solely from function values, and the global Lipschitz modulus is found when it exists. Some examples of classes of globally Lipschitz continuous convex functions beside the norms are also provided along with their global Lipschitz modulus.

   @article{lipschitz,
     title = {Lipschitz Modulus of Convex Functions via Function Values},
     author = {Khanh, P. D. and Khoa, V. V. H. and Phat, V. T. and Viet, L. D.},
     journal = {Optimization Letters},
     year = {2025},
     url = {https://link.springer.com/article/10.1007/s11590-025-02218-0},
     dimensions = {true},
   }