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|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.
|Electronic Publication Date:
|Moolekamp, Fred, Melchior, Peter. (2018). Block-simultaneous direction method of multipliers: a proximal primal-dual splitting algorithm for nonconvex problems with multiple constraints. OPTIMIZATION AND ENGINEERING, 19 (871 - 885. doi:10.1007/s11081-018-9380-y
|871 - 885
|Type of Material:
|OPTIMIZATION AND ENGINEERING
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