1,035 research outputs found
A successive difference-of-convex approximation method for a class of nonconvex nonsmooth optimization problems
We consider a class of nonconvex nonsmooth optimization problems whose
objective is the sum of a smooth function and a finite number of nonnegative
proper closed possibly nonsmooth functions (whose proximal mappings are easy to
compute), some of which are further composed with linear maps. This kind of
problems arises naturally in various applications when different regularizers
are introduced for inducing simultaneous structures in the solutions. Solving
these problems, however, can be challenging because of the coupled nonsmooth
functions: the corresponding proximal mapping can be hard to compute so that
standard first-order methods such as the proximal gradient algorithm cannot be
applied efficiently. In this paper, we propose a successive
difference-of-convex approximation method for solving this kind of problems. In
this algorithm, we approximate the nonsmooth functions by their Moreau
envelopes in each iteration. Making use of the simple observation that Moreau
envelopes of nonnegative proper closed functions are continuous {\em
difference-of-convex} functions, we can then approximately minimize the
approximation function by first-order methods with suitable majorization
techniques. These first-order methods can be implemented efficiently thanks to
the fact that the proximal mapping of {\em each} nonsmooth function is easy to
compute. Under suitable assumptions, we prove that the sequence generated by
our method is bounded and any accumulation point is a stationary point of the
objective. We also discuss how our method can be applied to concrete
applications such as nonconvex fused regularized optimization problems and
simultaneously structured matrix optimization problems, and illustrate the
performance numerically for these two specific applications
Super-Linear Convergence of Dual Augmented-Lagrangian Algorithm for Sparsity Regularized Estimation
We analyze the convergence behaviour of a recently proposed algorithm for
regularized estimation called Dual Augmented Lagrangian (DAL). Our analysis is
based on a new interpretation of DAL as a proximal minimization algorithm. We
theoretically show under some conditions that DAL converges super-linearly in a
non-asymptotic and global sense. Due to a special modelling of sparse
estimation problems in the context of machine learning, the assumptions we make
are milder and more natural than those made in conventional analysis of
augmented Lagrangian algorithms. In addition, the new interpretation enables us
to generalize DAL to wide varieties of sparse estimation problems. We
experimentally confirm our analysis in a large scale -regularized
logistic regression problem and extensively compare the efficiency of DAL
algorithm to previously proposed algorithms on both synthetic and benchmark
datasets.Comment: 51 pages, 9 figure
A quasi-Newton proximal splitting method
A new result in convex analysis on the calculation of proximity operators in
certain scaled norms is derived. We describe efficient implementations of the
proximity calculation for a useful class of functions; the implementations
exploit the piece-wise linear nature of the dual problem. The second part of
the paper applies the previous result to acceleration of convex minimization
problems, and leads to an elegant quasi-Newton method. The optimization method
compares favorably against state-of-the-art alternatives. The algorithm has
extensive applications including signal processing, sparse recovery and machine
learning and classification
Forward-backward truncated Newton methods for convex composite optimization
This paper proposes two proximal Newton-CG methods for convex nonsmooth
optimization problems in composite form. The algorithms are based on a a
reformulation of the original nonsmooth problem as the unconstrained
minimization of a continuously differentiable function, namely the
forward-backward envelope (FBE). The first algorithm is based on a standard
line search strategy, whereas the second one combines the global efficiency
estimates of the corresponding first-order methods, while achieving fast
asymptotic convergence rates. Furthermore, they are computationally attractive
since each Newton iteration requires the approximate solution of a linear
system of usually small dimension
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