arXiv subjects

V. S. T. Long

Publications and source records attributed to V. S. T. Long.

6 recordsLinked to original sources

arXiv2026

Stability Analysis of Generalized Multi-Source Weber Problems

This paper presents a stability analysis of the generalized multi-source Weber problems under the influence of data perturbation within the framework of the Minkowski function. First, we establish explicit Lipschitz constants for the objective functions and optimal value functions. Then, we prove several properties of the global optimal solution sets. Furthermore, we provide sufficient conditions for the upper semicontinuity of global solution mappings and the inner semicontinuity of local solution mappings. Three illustrative examples are constructed. Our results give indirect answers to several open questions regarding the stability of local optimal solution mappings of the optimization problems discussed.

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arXiv2026

On the Strong Quasiconvexity of Norms and Distance Functions

This paper studies the strong quasiconvexity of norm and distance functions in finite-dimensional normed spaces. Although the Euclidean norm is known to be strongly quasiconvex on bounded convex sets, a complete characterization of this property for general norms remains open. We establish necessary and sufficient conditions for a norm function to be strongly quasiconvex on a convex set. We also initiate the study of the strong quasiconvexity of distance functions. Our results provide new insights into the geometric properties of norm and distance functions and extend several existing results in the literature.

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arXiv2026

The Existence and Stability of Generalized Multi-Source Weber Problems

This paper studies the generalized multi-source Weber problem with set-valued targets in the framework of minimal time functions. We first establish the existence of global and local optimal solutions and investigate several qualitative properties of the corresponding solution sets, including closedness, compactness, and conditions ensuring boundedness or unboundedness. Next, we derive explicit Lipschitz continuity properties for the objective function and the associated optimal value function with respect to perturbations of the target sets. We then introduce the global and local solution mappings and study their stability properties from the viewpoint of set-valued analysis. These results provide a quantitative and qualitative sensitivity analysis for the generalized multi-source Weber problem in the setting of minimal time functions.

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arXiv2026

A New Notion of Tykhonov Well-Posedness for Optimization Problems

Building upon the minimal time function, we propose and study a novel notion of Tykhonov well-posedness with respect to a set of directions for optimization problems. This concept generalizes the classical Tykhonov well-posedness by focusing on existence, stability and convergence along specific directions, rather than over the entire space. We first establish several characterizations of Tykhonov well-posedness with respect to a set of directions, formulated in terms of the diameter of level sets and admissible functions. We then investigate relationships between these level sets and admissible functions. To highlight the advantages of the proposed framework, we present several illustrative examples. In particular, we show that by selecting a suitable set of directions, optimization problems that are not well-posed in the classical sense may still be Tykhonov well-posed with respect to those directions. This viewpoint not only broadens the theoretical landscape of well-posedness but also has practical implications, as it allows numerical methods to be effectively adapted so that the generated sequences converge reliably to minimizers.

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arXiv2026

Solving Regularized Multifacility Location Problems with Unknown Number of Centers via Difference-of-Convex Optimization

In this paper, we develop optimization methods for a new model of multifacility location problems defined by a Minkowski gauge with Laplace-type regularization terms. The model is analyzed from both theoretical and numerical perspectives. In particular, we establish the existence of optimal solutions and study qualitative properties of global minimizers. By combining Nesterov's smoothing technique with recent advances in difference-of-convex optimization, following the pioneering work of P. D. Tao and L. T. H. An and others, we propose efficient numerical algorithms for minimizing the objective function of this model. As an application, our approach provides an effective method for determining the number of centers in gauge-based multifacility location and clustering problems. Our results extend and complement recent developments.

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arXiv2025

Qualitative and Generalized Differentiation Properties of Optimal Value Functions with Applications to Duality

This paper investigates general and generalized differentiation properties of the optimal value function associated with perturbed optimization problems. Fundamental results on nearly convex sets and functions in infinite-dimensional spaces are then established. We proceed by analyzing general properties of the optimal value function, including its domain, epigraph, strict epigraph, near convexity, semicontinuity, and Lipschitz-type continuity in both convex and nonconvex settings. Subsequently, we derive calculus rules and representation formulas for the $ε$-subdifferentials of the optimal value function and its Fenchel conjugate. We then develop a duality framework for constrained optimization problems with set-valued constraints using the Fenchel conjugate for set-valued mappings. This approach provides new perspectives on duality in generalized settings.

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