1,267 research outputs found
On iterated minimization in nonconvex optimization
In dynamic programming and decomposition methods one often applies an iterated minimization procedure. The problem variables are partitioned into several blocks, say x and y. Treating y as a parameter, the first phase consists of minimization with respect to the variable x. In a second phase the minimization of the resulting optimal value function depending on y is considered. In this paper we treat this basic idea on a local level. It turns out that strong stability (in the sense of Kojima) in the first phase is a natural assumption. In order to show that the iterated local minima of the parametric problem lead to a local minimum for the whole problem, we use a generalized version of a positive definiteness criterion of Fujiwara-Han-Mangasarian
Weighted Schatten -Norm Minimization for Image Denoising and Background Subtraction
Low rank matrix approximation (LRMA), which aims to recover the underlying
low rank matrix from its degraded observation, has a wide range of applications
in computer vision. The latest LRMA methods resort to using the nuclear norm
minimization (NNM) as a convex relaxation of the nonconvex rank minimization.
However, NNM tends to over-shrink the rank components and treats the different
rank components equally, limiting its flexibility in practical applications. We
propose a more flexible model, namely the Weighted Schatten -Norm
Minimization (WSNM), to generalize the NNM to the Schatten -norm
minimization with weights assigned to different singular values. The proposed
WSNM not only gives better approximation to the original low-rank assumption,
but also considers the importance of different rank components. We analyze the
solution of WSNM and prove that, under certain weights permutation, WSNM can be
equivalently transformed into independent non-convex -norm subproblems,
whose global optimum can be efficiently solved by generalized iterated
shrinkage algorithm. We apply WSNM to typical low-level vision problems, e.g.,
image denoising and background subtraction. Extensive experimental results
show, both qualitatively and quantitatively, that the proposed WSNM can more
effectively remove noise, and model complex and dynamic scenes compared with
state-of-the-art methods.Comment: 13 pages, 11 figure
On the monotone and primal-dual active set schemes for -type problems,
Nonsmooth nonconvex optimization problems involving the quasi-norm,
, of a linear map are considered. A monotonically convergent
scheme for a regularized version of the original problem is developed and
necessary optimality conditions for the original problem in the form of a
complementary system amenable for computation are given. Then an algorithm for
solving the above mentioned necessary optimality conditions is proposed. It is
based on a combination of the monotone scheme and a primal-dual active set
strategy. The performance of the two algorithms is studied by means of a series
of numerical tests in different cases, including optimal control problems,
fracture mechanics and microscopy image reconstruction
Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection
A number of variable selection methods have been proposed involving nonconvex
penalty functions. These methods, which include the smoothly clipped absolute
deviation (SCAD) penalty and the minimax concave penalty (MCP), have been
demonstrated to have attractive theoretical properties, but model fitting is
not a straightforward task, and the resulting solutions may be unstable. Here,
we demonstrate the potential of coordinate descent algorithms for fitting these
models, establishing theoretical convergence properties and demonstrating that
they are significantly faster than competing approaches. In addition, we
demonstrate the utility of convexity diagnostics to determine regions of the
parameter space in which the objective function is locally convex, even though
the penalty is not. Our simulation study and data examples indicate that
nonconvex penalties like MCP and SCAD are worthwhile alternatives to the lasso
in many applications. In particular, our numerical results suggest that MCP is
the preferred approach among the three methods.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS388 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Efficient Regret Minimization in Non-Convex Games
We consider regret minimization in repeated games with non-convex loss
functions. Minimizing the standard notion of regret is computationally
intractable. Thus, we define a natural notion of regret which permits efficient
optimization and generalizes offline guarantees for convergence to an
approximate local optimum. We give gradient-based methods that achieve optimal
regret, which in turn guarantee convergence to equilibrium in this framework.Comment: Published as a conference paper at ICML 201
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