1,163 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 Smooth Primal-Dual Optimization Framework for Nonsmooth Composite Convex Minimization
We propose a new first-order primal-dual optimization framework for a convex
optimization template with broad applications. Our optimization algorithms
feature optimal convergence guarantees under a variety of common structure
assumptions on the problem template. Our analysis relies on a novel combination
of three classic ideas applied to the primal-dual gap function: smoothing,
acceleration, and homotopy. The algorithms due to the new approach achieve the
best known convergence rate results, in particular when the template consists
of only non-smooth functions. We also outline a restart strategy for the
acceleration to significantly enhance the practical performance. We demonstrate
relations with the augmented Lagrangian method and show how to exploit the
strongly convex objectives with rigorous convergence rate guarantees. We
provide numerical evidence with two examples and illustrate that the new
methods can outperform the state-of-the-art, including Chambolle-Pock, and the
alternating direction method-of-multipliers algorithms.Comment: 35 pages, accepted for publication on SIAM J. Optimization. Tech.
Report, Oct. 2015 (last update Sept. 2016
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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