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Largest Volume Inscribed Rectangles in Convex Sets Defined by Finite Number of Inequalities

Author

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  • Mehdi Behroozi

    (Department of Mechanical and Industrial Engineering, Northeastern University, Boston, Massachusetts 02115)

Abstract

This paper considers the problem of finding maximum volume (axis-aligned) inscribed boxes in a compact convex set, defined by a finite number of convex inequalities, and presents optimization and geometric approaches for solving them. Several optimization models are developed that can be easily generalized to find other inscribed geometric shapes such as triangles, rhombi, and squares. To find the largest volume axis-aligned inscribed rectangles in the higher dimensions, an interior-point method algorithm is presented and analyzed. For two-dimensional space, a parametrized optimization approach is developed to find the largest area (axis-aligned) inscribed rectangles in convex sets. The optimization approach provides a uniform framework for solving a wide variety of relevant problems. Finally, two computational geometric ( 1 − ε ) –approximation algorithms with sublinear running times are presented that improve the previous results.

Suggested Citation

  • Mehdi Behroozi, 2024. "Largest Volume Inscribed Rectangles in Convex Sets Defined by Finite Number of Inequalities," INFORMS Journal on Computing, INFORMS, vol. 36(3), pages 787-819, May.
    • Handle: RePEc:inm:orijoc:v:36:y:2024:i:3:p:787-819
    DOI: 10.1287/ijoc.2022.0239
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    References listed on IDEAS

    as
    1. John Gunnar Carlsson & Fan Jia & Ying Li, 2014. "An Approximation Algorithm for the Continuous k -Medians Problem in a Convex Polygon," INFORMS Journal on Computing, INFORMS, vol. 26(2), pages 280-289, May.
    2. John Carlsson & Mehdi Behroozi & Xiang Li, 2016. "Geometric partitioning and robust ad-hoc network design," Annals of Operations Research, Springer, vol. 238(1), pages 41-68, March.
    3. John Gunnar Carlsson & Mehdi Behroozi & Xiang Li, 2016. "Geometric partitioning and robust ad-hoc network design," Annals of Operations Research, Springer, vol. 238(1), pages 41-68, March.
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