An unconstrained minimization method for solving low-rank SDP relaxations of the maxcut problem

2009 ◽  
Vol 126 (1) ◽  
pp. 119-146 ◽  
Author(s):  
Luigi Grippo ◽  
Laura Palagi ◽  
Veronica Piccialli
Keyword(s):  
Unconstrained Minimization ◽  
Low Rank ◽  
Minimization Method ◽  
Maxcut Problem

scholarly journals An Active Set Smoothing Method for Solving Unconstrained Minimax Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-25
Author(s):  
Zhengyong Zhou ◽  
Qi Yang
Keyword(s):  
Minimax Problems ◽  
Unconstrained Minimization ◽  
Smoothing Method ◽  
Smoothing Function ◽  
Active Set ◽  
Maximum Function ◽  
Minimization Method ◽  
Lipschitz Continuous ◽  
Locally Lipschitz ◽  
Smoothing Parameters

In this paper, an active set smoothing function based on the plus function is constructed for the maximum function. The active set strategy used in the smoothing function reduces the number of gradients and Hessians evaluations of the component functions in the optimization. Combing the active set smoothing function, a simple adjustment rule for the smoothing parameters, and an unconstrained minimization method, an active set smoothing method is proposed for solving unconstrained minimax problems. The active set smoothing function is continuously differentiable, and its gradient is locally Lipschitz continuous and strongly semismooth. Under the boundedness assumption on the level set of the objective function, the convergence of the proposed method is established. Numerical experiments show that the proposed method is feasible and efficient, particularly for the minimax problems with very many component functions.

Minimization of the difference of Nuclear and Frobenius norms for noisy low rank matrix recovery

2019 ◽  
Vol 18 (02) ◽  
pp. 1950056
Author(s):  
Yun Cai
Keyword(s):  
Linear Measurement ◽  
Low Rank ◽  
Bounded Noise ◽  
Linear Measurements ◽  
Minimization Method ◽  
Matrix Recovery ◽  
Rank Matrix ◽  
The Difference ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix

This paper considers recovery of matrices that are low rank or approximately low rank from linear measurements corrupted with additive noise. We study minimization of the difference of Nuclear and Frobenius norms (abbreviated as [Formula: see text] norm) as a nonconvex and Lipschitz continuous metric for solving this noisy low rank matrix recovery problem. We mainly study two types of bounded observation noisy low rank matrix recovery problems, including the [Formula: see text]-norm bounded noise and the Dantizg Selector noise. Based on the matrix restricted isometry property (abbreviated as M-RIP), we prove that this [Formula: see text] norm-based minimization method can stably recover a (approximately) low rank matrix in the two types bounded noisy low rank matrix recovery problems. In addition, we use the truncated difference of Nuclear and Frobenius norms (denoted as the truncated [Formula: see text] norm) to recover a low rank matrix when the observation noise is the Dantizg Selector noise. We give the stable recovery result for this truncated [Formula: see text] norm minimization in Dantizg Selector noise case when the linear measurement map satisfies the M-RIP condition.

scholarly journals Image Denoising Using Sparsifying Transform Learning and Weighted Singular Values Minimization

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Yanwei Zhao ◽  
Ping Yang ◽  
Qiu Guan ◽  
Jianwei Zheng ◽  
Wanliang Wang
Keyword(s):  
Image Denoising ◽  
Convex Relaxation ◽  
Singular Values ◽  
Low Rank ◽  
Transform Domain ◽  
Minimization Method ◽  
Acceptable Solution ◽  
Image Domain ◽  
Image Prior ◽  
Efficient Alternative

In image denoising (IDN) processing, the low-rank property is usually considered as an important image prior. As a convex relaxation approximation of low rank, nuclear norm-based algorithms and their variants have attracted a significant attention. These algorithms can be collectively called image domain-based methods whose common drawback is the requirement of great number of iterations for some acceptable solution. Meanwhile, the sparsity of images in a certain transform domain has also been exploited in image denoising problems. Sparsity transform learning algorithms can achieve extremely fast computations as well as desirable performance. By taking both advantages of image domain and transform domain in a general framework, we propose a sparsifying transform learning and weighted singular values minimization method (STLWSM) for IDN problems. The proposed method can make full use of the preponderance of both domains. For solving the nonconvex cost function, we also present an efficient alternative solution for acceleration. Experimental results show that the proposed STLWSM achieves improvement both visually and quantitatively with a large margin over state-of-the-art approaches based on an alternatively single domain. It also needs much less iteration than all the image domain algorithms.

scholarly journals Projection Free Rank-Drop Steps

2017 ◽  
Author(s):  
Edward Cheung ◽  
Yuying Li
Keyword(s):  
Optimization Problems ◽  
High Rank ◽  
Nuclear Norm ◽  
Low Rank ◽  
Common Variants ◽  
Time And Space ◽  
Minimization Method ◽  
Drop Method ◽  
Constrained Problems ◽  
Free Rank

The Frank-Wolfe (FW) algorithm has been widely used in solving nuclear norm constrained problems, since it does not require projections. However, FW often yields high rank intermediate iterates, which can be very expensive in time and space costs for large problems. To address this issue, we propose a rank-drop method for nuclear norm constrained problems. The goal is to generate descent steps that lead to rank decreases, maintaining low-rank solutions throughout the algorithm. Moreover, the optimization problems are constrained to ensure that the rank-drop step is also feasible and can be readily incorporated into a projection-free minimization method, e.g., Frank-Wolfe. We demonstrate that by incorporating rank-drop steps into the Frank-Wolfe algorithm, the rank of the solution is greatly reduced compared to the original Frank-Wolfe or its common variants.

Crosstalk Minimization Method for Eye-tracking-based 3D Display

2020 ◽  
Author(s):  
Seok Lee ◽  
Juyong Park ◽  
Dongkyung Nam
Keyword(s):  
Image Processing ◽  
Eye Tracking ◽  
Three Dimensional ◽  
Processing Method ◽  
Eye Position ◽  
Relative Location ◽  
3D Display ◽  
Minimization Method ◽  
3D Crosstalk ◽  
Pixel Value

In this article, the authors present an image processing method to reduce three-dimensional (3D) crosstalk for eye-tracking-based 3D display. Specifically, they considered 3D pixel crosstalk and offset crosstalk and applied different approaches based on its characteristics. For 3D pixel crosstalk which depends on the viewer’s relative location, they proposed output pixel value weighting scheme based on viewer’s eye position, and for offset crosstalk they subtracted luminance of crosstalk components according to the measured display crosstalk level in advance. By simulations and experiments using the 3D display prototypes, the authors evaluated the effectiveness of proposed method.

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