### Abstract

Many interesting and fundamentally practical optimization problems, ranging from optics, to signal processing, to radar and acoustics, involve constraints on the Fourier transform of a function. It is well-known that the fast Fourier transform (fft) is a recursive algorithm that can dramatically improve the efficiency for computing the discrete Fourier transform. However, because it is recursive, it is difficult to embed into a linear optimization problem. In this paper, we explain the main idea behind the fast Fourier transform and show how to adapt it in such a manner as to make it encodable as constraints in an optimization problem. We demonstrate a real-world problem from the field of high-contrast imaging. On this problem, dramatic improvements are translated to an ability to solve problems with a much finer grid of discretized points. As we shall show, in general, the "fast Fourier" version of the optimization constraints produces a larger but sparser constraint matrix and therefore one can think of the fast Fourier transform as a method of sparsifying the constraints in an optimization problem, which is usually a good thing.

Original language | English (US) |
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Pages (from-to) | 53-69 |

Number of pages | 17 |

Journal | Mathematical Programming Computation |

Volume | 4 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2012 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Software

### Keywords

- Cooley-Tukey algorithm
- Fast Fourier transform (fft)
- Fourier transform
- High-contrast imaging
- Interior-point methods
- Linear programming
- Optimization