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

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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 \(O\left( \frac{1}{\sqrt{\epsilon }}\ln (\frac{1}{\epsilon })\right) \) complexity bound is better than the worst-case complexity bound of \(O(\frac{1}{\epsilon })\) of Gilbert’s algorithm.

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Acknowledgements

This article was supported by the National Natural Science Foundation of China under Grant Nos. 11401152 and 11950410503. Research of the second author was supported by the China Postdoctoral Science Foundation under Grant No. 2017M622991 and the Vietnam National Foundation for Science and Technology Development under Grant No. 101.01-2017.325. The authors would like to thank anonymous reviewers for insightful comments that helped to greatly improve the manuscript.

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  1. Department of Mathematics, Hangzhou Normal University, Hangzhou, China

    Xiaolong Qin

  2. Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Xiaolong Qin & Nguyen Thai An

  3. Institute of Research and Development, Duy Tan University, Danang, Vietnam

    Nguyen Thai An

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  1. Xiaolong Qin
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  2. Nguyen Thai An
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Qin, X., An, N.T. Smoothing algorithms for computing the projection onto a Minkowski sum of convex sets. Comput Optim Appl 74, 821–850 (2019). https://doi.org/10.1007/s10589-019-00124-7

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  • DOI: https://doi.org/10.1007/s10589-019-00124-7

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