1,098 research outputs found
A proximal minimization algorithm for structured nonconvex and nonsmooth problems
We propose a proximal algorithm for minimizing objective functions consisting
of three summands: the composition of a nonsmooth function with a linear
operator, another nonsmooth function, each of the nonsmooth summands depending
on an independent block variable, and a smooth function which couples the two
block variables. The algorithm is a full splitting method, which means that the
nonsmooth functions are processed via their proximal operators, the smooth
function via gradient steps, and the linear operator via matrix times vector
multiplication. We provide sufficient conditions for the boundedness of the
generated sequence and prove that any cluster point of the latter is a KKT
point of the minimization problem. In the setting of the Kurdyka-\L{}ojasiewicz
property we show global convergence, and derive convergence rates for the
iterates in terms of the \L{}ojasiewicz exponent
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
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