2 research outputs found
A Smoothing SQP Framework for a Class of Composite Minimization over Polyhedron
The composite minimization problem over a general polyhedron
has received various applications in machine learning, wireless communications,
image restoration, signal reconstruction, etc. This paper aims to provide a
theoretical study on this problem. Firstly, we show that for any fixed ,
finding the global minimizer of the problem, even its unconstrained
counterpart, is strongly NP-hard. Secondly, we derive Karush-Kuhn-Tucker (KKT)
optimality conditions for local minimizers of the problem. Thirdly, we propose
a smoothing sequential quadratic programming framework for solving this
problem. The framework requires a (approximate) solution of a convex quadratic
program at each iteration. Finally, we analyze the worst-case iteration
complexity of the framework for returning an -KKT point; i.e., a
feasible point that satisfies a perturbed version of the derived KKT optimality
conditions. To the best of our knowledge, the proposed framework is the first
one with a worst-case iteration complexity guarantee for solving composite
minimization over a general polyhedron
Effective two-stage image segmentation: a new non-Lipschitz decomposition approach with convergent algorithm
Image segmentation is an important median level vision topic. Accurate and
efficient multiphase segmentation for images with intensity inhomogeneity is
still a great challenge. We present a new two-stage multiphase segmentation
method trying to tackle this, where the key is to compute an inhomogeneity-free
approximate image. For this, we propose to use a new non-Lipschitz variational
decomposition model in the first stage. The minimization problem is solved by
an iterative support shrinking algorithm, with a global convergence guarantee
and a lower bound theory of the image gradient of the iterative sequence. The
latter indicates that the generated approximate image (inhomogeneity-corrected
component) is with very neat edges and suitable for the following thresholding
operation. In the second stage, the segmentation is done by applying a
widely-used simple thresholding technique to the piecewise constant
approximation. Numerical experiments indicate good convergence properties and
effectiveness of our method in multiphase segmentation for either clean or
noisy homogeneous and inhomogeneous images. Both visual and quantitative
comparisons with some state-of-the-art approaches demonstrate the performance
advantages of our non-Lipschitz based method