1,945 research outputs found
Convex optimization over intersection of simple sets: improved convergence rate guarantees via an exact penalty approach
We consider the problem of minimizing a convex function over the intersection
of finitely many simple sets which are easy to project onto. This is an
important problem arising in various domains such as machine learning. The main
difficulty lies in finding the projection of a point in the intersection of
many sets. Existing approaches yield an infeasible point with an
iteration-complexity of for nonsmooth problems with no
guarantees on the in-feasibility. By reformulating the problem through exact
penalty functions, we derive first-order algorithms which not only guarantees
that the distance to the intersection is small but also improve the complexity
to and for smooth functions. For
composite and smooth problems, this is achieved through a saddle-point
reformulation where the proximal operators required by the primal-dual
algorithms can be computed in closed form. We illustrate the benefits of our
approach on a graph transduction problem and on graph matching
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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