### Abstract

We introduce a generalization of the linearized Alternating Direction Method of Multipliers to optimize a real-valued function f of multiple arguments with potentially multiple constraints g_{∘} on each of them. The function f may be nonconvex as long as it is convex in every argument, while the constraints g_{∘} need to be convex but not smooth. If f is smooth, the proposed Block-Simultaneous Direction Method of Multipliers (bSDMM) can be interpreted as a proximal analog to inexact coordinate descent methods under constraints. Unlike alternative approaches for joint solvers of multiple-constraint problems, we do not require linear operators L of a constraint function g(L·) to be invertible or linked between each other. bSDMM is well-suited for a range of optimization problems, in particular for data analysis, where f is the likelihood function of a model and L could be a transformation matrix describing e.g. finite differences or basis transforms. We apply bSDMM to the Non-negative Matrix Factorization task of a hyperspectral unmixing problem and demonstrate convergence and effectiveness of multiple constraints on both matrix factors. The algorithms are implemented in python and released as an open-source package.

Original language | English (US) |
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Pages (from-to) | 871-885 |

Number of pages | 15 |

Journal | Optimization and Engineering |

Volume | 19 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2018 |

### All Science Journal Classification (ASJC) codes

- Software
- Civil and Structural Engineering
- Aerospace Engineering
- Mechanical Engineering
- Control and Optimization
- Electrical and Electronic Engineering

### Keywords

- Block coordinate descent
- Non-negative matrix factorization
- Nonconvex optimization
- Optimization
- Proximal algorithms