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Block-simultaneous direction method of multipliers: a proximal primal-dual splitting algorithm for nonconvex problems with multiple constraints

Moolekamp, Fred / Melchior, Peter

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 $g_{\circ}$ on each of them. The function f may be nonconvex as long as it is convex in every argument, while the constraints $g_{\circ}$ 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 $g (L \cdot)$ 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.

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Block-simultaneous direction method of multipliers: a proximal primal-dual splitting algorithm for nonconvex problems with multiple constraints

Moolekamp, Fred / Melchior, Peter

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 $g_{\circ}$ on each of them. The function f may be nonconvex as long as it is convex in every argument, while the constraints $g_{\circ}$ 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 $g (L \cdot)$ 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.

Block-simultaneous direction method of multipliers: a proximal primal-dual splitting algorithm for nonconvex problems with multiple constraints

Moolekamp, Fred / Melchior, Peter

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 $g_{\circ}$ on each of them. The function f may be nonconvex as long as it is convex in every argument, while the constraints $g_{\circ}$ 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 $g (L \cdot)$ 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.

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Title:

Block-simultaneous direction method of multipliers: a proximal primal-dual splitting algorithm for nonconvex problems with multiple constraints

Contributors:

Moolekamp, Fred (author) / Melchior, Peter (author)

Published in:

Optimization and Engineering ; 19 ; 871-885

Publication date:

2018-03-20

Size:

15 pages

ISSN:

1573-2924 , 1389-4420

DOI:

https://doi.org/10.1007/s11081-018-9380-y

Type of media:

Article (Journal)

Type of material:

Electronic Resource

Language:

English

Keywords:

Optimization , Proximal algorithms , Nonconvex optimization , Block coordinate descent , Non-negative matrix factorization Mathematics , Optimization , Engineering, general , Systems Theory, Control , Environmental Management , Operations Research/Decision Theory , Financial Engineering

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