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On the Convergence of Alternating Direction Lagrangian Methods for Nonconvex Structured Optimization Problems
Nonconvex and structured optimization problems arise in many engineering
applications that demand scalable and distributed solution methods. The study
of the convergence properties of these methods is in general difficult due to
the nonconvexity of the problem. In this paper, two distributed solution
methods that combine the fast convergence properties of augmented
Lagrangian-based methods with the separability properties of alternating
optimization are investigated. The first method is adapted from the classic
quadratic penalty function method and is called the Alternating Direction
Penalty Method (ADPM). Unlike the original quadratic penalty function method,
in which single-step optimizations are adopted, ADPM uses an alternating
optimization, which in turn makes it scalable. The second method is the
well-known Alternating Direction Method of Multipliers (ADMM). It is shown that
ADPM for nonconvex problems asymptotically converges to a primal feasible point
under mild conditions and an additional condition ensuring that it
asymptotically reaches the standard first order necessary conditions for local
optimality are introduced. In the case of the ADMM, novel sufficient conditions
under which the algorithm asymptotically reaches the standard first order
necessary conditions are established. Based on this, complete convergence of
ADMM for a class of low dimensional problems are characterized. Finally, the
results are illustrated by applying ADPM and ADMM to a nonconvex localization
problem in wireless sensor networks.Comment: 13 pages, 6 figure
Nonconvex Generalization of ADMM for Nonlinear Equality Constrained Problems
The ever-increasing demand for efficient and distributed optimization
algorithms for large-scale data has led to the growing popularity of the
Alternating Direction Method of Multipliers (ADMM). However, although the use
of ADMM to solve linear equality constrained problems is well understood, we
lacks a generic framework for solving problems with nonlinear equality
constraints, which are common in practical applications (e.g., spherical
constraints). To address this problem, we are proposing a new generic ADMM
framework for handling nonlinear equality constraints, neADMM. After
introducing the generalized problem formulation and the neADMM algorithm, the
convergence properties of neADMM are discussed, along with its sublinear
convergence rate , where is the number of iterations. Next, two
important applications of neADMM are considered and the paper concludes by
describing extensive experiments on several synthetic and real-world datasets
to demonstrate the convergence and effectiveness of neADMM compared to existing
state-of-the-art methods
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