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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.11555 |
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| _version_ | 1866914260150386688 |
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| author | Nath, Triloki Choudhary, Manohar Pandey, Ram K. |
| author_facet | Nath, Triloki Choudhary, Manohar Pandey, Ram K. |
| contents | This paper presents a new extension of the classical Heron problem, termed the generalized $(k,m)$-Heron problem, which seeks an optimal configuration among $k$ feasible and $m$ target non-empty closed convex sets in $\mathbb{R}^n$. The problem is formulated as finding a point in each set that minimizes the pairwise distances from the points in the $k$-feasible sets to the points in the $m$-target sets. This formulation leads to a convex optimization framework that generalizes several well-known geometric distance problems. Using tools from convex analysis, we establish fundamental results on existence, uniqueness, and first-order optimality conditions through subdifferential calculus and normal cone theory. Building on these insights, a Projected Subgradient Algorithm (PSA) is proposed for numerical solution, and its convergence is rigorously proved under a diminishing step-size rule. Numerical experiments in $\mathbb{R}^2$ and $\mathbb{R}^3$ illustrate the algorithm's stability, geometric accuracy, and computational efficiency. Overall, this work provides a comprehensive analytical and algorithmic framework for multi-set geometric optimization with promising implications for location science, robotics, and computational geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_11555 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Generalized $(k,m)$ Heron Problem:Optimality Conditions and Algorithm Nath, Triloki Choudhary, Manohar Pandey, Ram K. Optimization and Control 90B85, 51M04 This paper presents a new extension of the classical Heron problem, termed the generalized $(k,m)$-Heron problem, which seeks an optimal configuration among $k$ feasible and $m$ target non-empty closed convex sets in $\mathbb{R}^n$. The problem is formulated as finding a point in each set that minimizes the pairwise distances from the points in the $k$-feasible sets to the points in the $m$-target sets. This formulation leads to a convex optimization framework that generalizes several well-known geometric distance problems. Using tools from convex analysis, we establish fundamental results on existence, uniqueness, and first-order optimality conditions through subdifferential calculus and normal cone theory. Building on these insights, a Projected Subgradient Algorithm (PSA) is proposed for numerical solution, and its convergence is rigorously proved under a diminishing step-size rule. Numerical experiments in $\mathbb{R}^2$ and $\mathbb{R}^3$ illustrate the algorithm's stability, geometric accuracy, and computational efficiency. Overall, this work provides a comprehensive analytical and algorithmic framework for multi-set geometric optimization with promising implications for location science, robotics, and computational geometry. |
| title | A Generalized $(k,m)$ Heron Problem:Optimality Conditions and Algorithm |
| topic | Optimization and Control 90B85, 51M04 |
| url | https://arxiv.org/abs/2601.11555 |