Abstract
In this paper, we propose new approximation algorithms for a NP-hard problem, i.e., weighted maximin dispersion problem. By using a uniformly distributed random sample method, we first propose a new random approximation algorithm for box constrained or ball constrained weighted maximin dispersion problems and analyze its approximation bound respectively. Moreover, we propose two improved approximation algorithms by combining our technique with an existing binary sample technique for both cases. To the best of our knowledge, they are the best approximation bounds for both box constrained and ball constrained weighted maximin dispersion problems respectively.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dang, C.D., Lan, G.: Randomized First-Order Methods for Saddle Point Optimization. Technical Report, Department of Industrial and Systems Engineering, University of Florida (2015)
Dasarthy, B., White, L.J.: A Maximin Location Problem. Oper. Res. 28(6), 1385–1401 (1980)
Haines, S., Loeppky, J., Tseng, P., Wang, X.: Convex Relaxations of The Weighted Maxmin Dispersion Problem. SIAM J. Optim. 23(4), 2264–2294 (2013)
Johbson, M.E., Moore, L.M., Ylvisaker, D.: Minimax and Maximin Distance Designs. Statist. Plann. J. 26(2), 131–148 (1990)
Lin, T., Jin, C., Jordan, M.: Near-optimal Algorithms for Minimax Optimization. Proceedings of Thirty Third Conference on Learning Theory, PMLR 125, 2738–2779 (2020)
Lu, S., Tsaknakis, I., Hong, M., Chen, Y.: Hybrid Block Successive Approximation for One-sided Non-convex Min-max Problems: algorithms and applications. IEEE Trans. Sign. Proc. 68, 3676–3691 (2020)
Pan, W., Shen, J., Xu, Z.: An Efficient Algorithm for Nonconvex-linear Minimax Optimization Problem and its Application in Solving Weighted Maximin Dispsersion Problem. Comput. Optim. Appl. 78(1), 287–306 (2021)
Ravi, S.S., Rosenkrantz, D.J., Tayi, G.K.: Heuristic and Special Case Algorithms for Dispersion Problems. Oper. Res. 42(2), 299–310 (1994)
Wang, S., Xia, Y.: On the Ball-constrained Weighted Maximin Dispersion Problem. SIAM J. Optim. 26(3), 1565–1588 (2016)
White, D.J.: A Heuristic Approach to A Weighted Maxmin Dispersion Problem. IMA J. Manag. Math. 7(3), 219–231 (1996)
Wu, Z.P., Xia, Y., Wang, S.: Approximating the Weighted Maximin Dispersion Problem Over an \(\ell _{p}-\)ball: SDP Relaxation is Misleading. Optim. Lett. 12, 875–883 (2018)
Xu, Z., Wang, S.W., Huang, J.J.: An Efficient Low Complexity Algorithm for Box-constrained Weighted Maximin Dispersion Problem. J. Ind. Manag. Optim. 17(2), 971–979 (2021)
Xu, Z., Zhang, H., Xu, Y., Lan, G.: A Unified Single-loop Alternating Gradient Projection Algorithm for Nonconvex-concave and Convex-nonconcave Minimax Problems. arXiv preprint arXiv:2006.02032, (2020)
Acknowledgements
We would like to thank the editor and two anonymous reviewers for their helpful comments and suggestions to improve this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Firdaus E. Udwadia.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research is supported by National Natural Science Foundation of China under the grant 12071279 and 11771208, and by General Project of Shanghai Natural Science Foundation (No. 20ZR1420600).
Rights and permissions
About this article
Cite this article
Wang, S., Xu, Z. New Approximation Algorithms for Weighted Maximin Dispersion Problem with Box or Ball Constraints. J Optim Theory Appl 190, 524–539 (2021). https://doi.org/10.1007/s10957-021-01893-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-021-01893-0