Newton-Type Alternating Minimization Algorithm for Convex Optimization

We propose a Newton-type alternating minimization algorithm (NAMA) for solving structured nonsmooth convex optimization problems where the sum of two functions is to be minimized, one being strongly convex and the other composed with a linear mapping. The proposed algorithm is a line-search method o...

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Vydáno v:IEEE transactions on automatic control Ročník 64; číslo 2; s. 697 - 711
Hlavní autoři: Stella, Lorenzo, Themelis, Andreas, Patrinos, Panagiotis
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York IEEE 01.02.2019
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:
Acceleration
Algorithms
Computational geometry
Continuity (mathematics)
Convergence
Convergence of numerical methods
Convex analysis
Convex functions
Convexity
Gradient methods
iterative methods
Mapping
Minimization
optimal control
Optimization
Penalty function
ISSN:0018-9286, 1558-2523
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Popis
Shrnutí:We propose a Newton-type alternating minimization algorithm (NAMA) for solving structured nonsmooth convex optimization problems where the sum of two functions is to be minimized, one being strongly convex and the other composed with a linear mapping. The proposed algorithm is a line-search method over a continuous, real-valued, exact penalty function for the corresponding dual problem, which is computed by evaluating the augmented Lagrangian at the primal points obtained by alternating minimizations. As a consequence, NAMA relies on exactly the same computations as the classical alternating minimization algorithm (AMA), also known as the dual-proximal gradient method. Under standard assumptions, the proposed algorithm converges with global sublinear and local linear rates, while under mild additional assumptions, the asymptotic convergence is superlinear, provided that the search directions are chosen according to quasi-Newton formulas. Due to its simplicity, the proposed method is well suited for embedded applications and large-scale problems. Experiments show that using limited-memory directions in NAMA greatly improves the convergence speed over AMA and its accelerated variant.
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ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2018.2872203