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

Author

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  • 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
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    References listed on IDEAS

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    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. 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.
    3. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    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.
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    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. 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.
    6. 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.
    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.

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