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

Xiaolong Qin () and Nguyen Thai An ()
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Xiaolong Qin: Hangzhou Normal University
Nguyen Thai An: University of Electronic Science and Technology of China

Computational Optimization and Applications, 2019, vol. 74, issue 3, No 9, 850 pages

Abstract: 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.

Keywords: Minimum norm problem; Minkowski sum of sets; Gilbert’s algorithm; Nesterov’s smoothing technique; Fast gradient method; SAGA; Primary 49J52; 49M29; Secondary 90C30 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (7)

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DOI: 10.1007/s10589-019-00124-7

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