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

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...

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Bibliographic Details
Published in:Computational optimization and applications 2019-12, Vol.74 (3), p.821-850
Main Authors: Qin, Xiaolong, An, Nguyen Thai
Format: Article
Language:English
Subjects:
Algorithms
Complexity
Computational geometry
Convex and Discrete Geometry
Convexity
Duality theorem
Management Science
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Path planning
Projection
Robotics
Smoothing
Statistics
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Summary: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 1 ϵ ln ( 1 ϵ ) complexity bound is better than the worst-case complexity bound of O ( 1 ϵ ) of Gilbert’s algorithm.
ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-019-00124-7