Abstract
In this paper we consider Newton’s problem of finding a convex body of least resistance. This problem could equivalently be written as a variational problem over concave functions in \({\mathbb {R}}^{2}\). We propose two different methods for solving it numerically. First, we discretize this problem by writing the concave solution function as a infimum over a finite number of affine functions. The discretized problem could be solved by standard optimization software efficiently. Second, we conjecture that the optimal body has a certain structure. We exploit this structure and obtain a variational problem in \({\mathbb {R}}^{1}\). Deriving its Euler–Lagrange equation yields a program with two unknowns, which can be solved quickly.
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Notes
All computations were done using a computer with two Intel Xeon Dual Core CPU (\(4 \times 3.0\,\text {GHz}\)) with \(16\,\text {GB}\) RAM.
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Wachsmuth, G. The numerical solution of Newton’s problem of least resistance. Math. Program. 147, 331–350 (2014). https://doi.org/10.1007/s10107-014-0756-2
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DOI: https://doi.org/10.1007/s10107-014-0756-2