IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v74y2019i3d10.1007_s10589-019-00124-7.html
   My bibliography  Save this article

Smoothing algorithms for computing the projection onto a Minkowski sum of convex sets

Author

Listed:
  • Xiaolong Qin

    (Hangzhou Normal University
    University of Electronic Science and Technology of China)

  • Nguyen Thai An

    (University of Electronic Science and Technology of China
    Duy Tan University)

Abstract

In this paper, the problem of computing the projection, and therefore the minimum distance, from a point onto a Minkowski sum of general convex sets is studied. Our approach is based on Nirenberg’s minimum norm duality theorem and Nesterov’s smoothing techniques. It is shown that the projection onto a Minkowski sum of sets can be represented as the sum of points on constituent sets so that, at these points, all of the sets share the same normal vector which is the negative of the dual solution. For numerically solving the problem, the most suitable algorithm is the one suggested by Gilbert (SIAM J Control 4:61–80, 1966). This algorithm has been widely used in collision detection and path planning in robotics. However, a main drawback of this method is that in some cases, it turns to be very slow as it approaches the solution. In this paper we proposed NESMINO whose $$O\left( \frac{1}{\sqrt{\epsilon }}\ln (\frac{1}{\epsilon })\right) $$O1ϵln(1ϵ) complexity bound is better than the worst-case complexity bound of $$O(\frac{1}{\epsilon })$$O(1ϵ) of Gilbert’s algorithm.

Suggested Citation

  • Xiaolong Qin & Nguyen Thai An, 2019. "Smoothing algorithms for computing the projection onto a Minkowski sum of convex sets," Computational Optimization and Applications, Springer, vol. 74(3), pages 821-850, December.
    • Handle: RePEc:spr:coopap:v:74:y:2019:i:3:d:10.1007_s10589-019-00124-7
    DOI: 10.1007/s10589-019-00124-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-019-00124-7
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10589-019-00124-7?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Boris Mordukhovich & Nguyen Nam, 2010. "Limiting subgradients of minimal time functions in Banach spaces," Journal of Global Optimization, Springer, vol. 46(4), pages 615-633, April.
    2. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Marguerite Frank & Philip Wolfe, 1956. "An algorithm for quadratic programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 95-110, March.
    4. Ming Yuan & Yi Lin, 2006. "Model selection and estimation in regression with grouped variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 49-67, February.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Lu-Chuan Ceng & Meijuan Shang, 2019. "Generalized Mann Viscosity Implicit Rules for Solving Systems of Variational Inequalities with Constraints of Variational Inclusions and Fixed Point Problems," Mathematics, MDPI, vol. 7(10), pages 1-18, October.
    2. Xiangfeng Wang & Junping Zhang & Wenxing Zhang, 2020. "The distance between convex sets with Minkowski sum structure: application to collision detection," Computational Optimization and Applications, Springer, vol. 77(2), pages 465-490, November.
    3. Chainarong Khunpanuk & Bancha Panyanak & Nuttapol Pakkaranang, 2022. "A New Construction and Convergence Analysis of Non-Monotonic Iterative Methods for Solving ρ -Demicontractive Fixed Point Problems and Variational Inequalities Involving Pseudomonotone Mapping," Mathematics, MDPI, vol. 10(4), pages 1-29, February.
    4. Lu-Chuan Ceng & Xiaolong Qin & Yekini Shehu & Jen-Chih Yao, 2019. "Mildly Inertial Subgradient Extragradient Method for Variational Inequalities Involving an Asymptotically Nonexpansive and Finitely Many Nonexpansive Mappings," Mathematics, MDPI, vol. 7(10), pages 1-19, September.
    5. Yinglin Luo & Meijuan Shang & Bing Tan, 2020. "A General Inertial Viscosity Type Method for Nonexpansive Mappings and Its Applications in Signal Processing," Mathematics, MDPI, vol. 8(2), pages 1-18, February.
    6. Bing Tan & Shanshan Xu & Songxiao Li, 2020. "Modified Inertial Hybrid and Shrinking Projection Algorithms for Solving Fixed Point Problems," Mathematics, MDPI, vol. 8(2), pages 1-12, February.
    7. Bing Tan & Zheng Zhou & Songxiao Li, 2020. "Strong Convergence of Modified Inertial Mann Algorithms for Nonexpansive Mappings," Mathematics, MDPI, vol. 8(4), pages 1-11, March.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Guillaume Sagnol & Edouard Pauwels, 2019. "An unexpected connection between Bayes A-optimal designs and the group lasso," Statistical Papers, Springer, vol. 60(2), pages 565-584, April.
    2. David Degras, 2021. "Sparse group fused lasso for model segmentation: a hybrid approach," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 15(3), pages 625-671, September.
    3. Silvia Villa & Lorenzo Rosasco & Sofia Mosci & Alessandro Verri, 2014. "Proximal methods for the latent group lasso penalty," Computational Optimization and Applications, Springer, vol. 58(2), pages 381-407, June.
    4. Ya-Feng Liu & Xin Liu & Shiqian Ma, 2019. "On the Nonergodic Convergence Rate of an Inexact Augmented Lagrangian Framework for Composite Convex Programming," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 632-650, May.
    5. Pi, J. & Wang, Honggang & Pardalos, Panos M., 2021. "A dual reformulation and solution framework for regularized convex clustering problems," European Journal of Operational Research, Elsevier, vol. 290(3), pages 844-856.
    6. Qihang Lin & Xi Chen & Javier Peña, 2014. "A sparsity preserving stochastic gradient methods for sparse regression," Computational Optimization and Applications, Springer, vol. 58(2), pages 455-482, June.
    7. Shipra Agrawal & Nikhil R. Devanur, 2019. "Bandits with Global Convex Constraints and Objective," Operations Research, INFORMS, vol. 67(5), pages 1486-1502, September.
    8. Daniel Dadush & László A. Végh & Giacomo Zambelli, 2020. "Rescaling Algorithms for Linear Conic Feasibility," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 732-754, May.
    9. Li, Peili & Xiao, Yunhai, 2018. "An efficient algorithm for sparse inverse covariance matrix estimation based on dual formulation," Computational Statistics & Data Analysis, Elsevier, vol. 128(C), pages 292-307.
    10. Tutz, Gerhard & Pößnecker, Wolfgang & Uhlmann, Lorenz, 2015. "Variable selection in general multinomial logit models," Computational Statistics & Data Analysis, Elsevier, vol. 82(C), pages 207-222.
    11. Jun Yan & Jian Huang, 2012. "Model Selection for Cox Models with Time-Varying Coefficients," Biometrics, The International Biometric Society, vol. 68(2), pages 419-428, June.
    12. Ye, Ya-Fen & Shao, Yuan-Hai & Deng, Nai-Yang & Li, Chun-Na & Hua, Xiang-Yu, 2017. "Robust Lp-norm least squares support vector regression with feature selection," Applied Mathematics and Computation, Elsevier, vol. 305(C), pages 32-52.
    13. Bakalli, Gaetan & Guerrier, Stéphane & Scaillet, Olivier, 2023. "A penalized two-pass regression to predict stock returns with time-varying risk premia," Journal of Econometrics, Elsevier, vol. 237(2).
    14. Shuichi Kawano, 2014. "Selection of tuning parameters in bridge regression models via Bayesian information criterion," Statistical Papers, Springer, vol. 55(4), pages 1207-1223, November.
    15. Peng, Heng & Lu, Ying, 2012. "Model selection in linear mixed effect models," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 109-129.
    16. Yize Zhao & Matthias Chung & Brent A. Johnson & Carlos S. Moreno & Qi Long, 2016. "Hierarchical Feature Selection Incorporating Known and Novel Biological Information: Identifying Genomic Features Related to Prostate Cancer Recurrence," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(516), pages 1427-1439, October.
    17. G. Aneiros & P. Vieu, 2016. "Sparse nonparametric model for regression with functional covariate," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(4), pages 839-859, October.
    18. Abdelfettah Laouzai & Rachid Ouafi, 2022. "A prediction model for atmospheric pollution reduction from urban traffic," Environment and Planning B, , vol. 49(2), pages 566-584, February.
    19. Wongsa-art, Pipat & Kim, Namhyun & Xia, Yingcun & Moscone, Francesco, 2024. "Varying coefficient panel data models and methods under correlated error components: Application to disparities in mental health services in England," Regional Science and Urban Economics, Elsevier, vol. 106(C).
    20. Dong, C. & Li, S., 2021. "Specification Lasso and an Application in Financial Markets," Cambridge Working Papers in Economics 2139, Faculty of Economics, University of Cambridge.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:coopap:v:74:y:2019:i:3:d:10.1007_s10589-019-00124-7. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.