In this paper we present some algorithms for minimization of DC function (difference of two convex functions). They are descent methods of the proximal-type which use the convex properties of the two convex functions ...In this paper we present some algorithms for minimization of DC function (difference of two convex functions). They are descent methods of the proximal-type which use the convex properties of the two convex functions separately. We also consider an approximate proximal point algorithm. Some properties of the ε-subdifferential and the ε-directional derivative are discussed. The convergence properties of the algorithms are established in both exact and approximate forms. Finally, we give some applications to the concave programming and maximum eigenvalue problems.展开更多
This paper presents a geometric characterization of convex sets in locally convex spaces onwhich a strong optimization theorem of the Stegall-type holds, and gives Collier's theorem ofw* Asplund spaces a localized...This paper presents a geometric characterization of convex sets in locally convex spaces onwhich a strong optimization theorem of the Stegall-type holds, and gives Collier's theorem ofw* Asplund spaces a localized setting.展开更多
Two-phase image segmentation is a fundamental task to partition an image into foreground and background.In this paper,two types of nonconvex and nonsmooth regularization models are proposed for basic two-phase segment...Two-phase image segmentation is a fundamental task to partition an image into foreground and background.In this paper,two types of nonconvex and nonsmooth regularization models are proposed for basic two-phase segmentation.They extend the convex regularization on the characteristic function on the image domain to the nonconvex case,which are able to better obtain piecewise constant regions with neat boundaries.By analyzing the proposed non-Lipschitz model,we combine the proximal alternating minimization framework with support shrinkage and linearization strategies to design our algorithm.This leads to two alternating strongly convex subproblems which can be easily solved.Similarly,we present an algorithm without support shrinkage operation for the nonconvex Lipschitz case.Using the Kurdyka-Lojasiewicz property of the objective function,we prove that the limit point of the generated sequence is a critical point of the original nonconvex nonsmooth problem.Numerical experiments and comparisons illustrate the effectiveness of our method in two-phase image segmentation.展开更多
In this paper, we derive an exact penalty function for nonconvex bilevel programming problem based on its KS form. Based on this exact penalty function a sufficient condition for KS to be partially calm is presented a...In this paper, we derive an exact penalty function for nonconvex bilevel programming problem based on its KS form. Based on this exact penalty function a sufficient condition for KS to be partially calm is presented and a necessary optimality condition for nonconvex bilevel programming problems is given. Some existing results about the differentiability of the value function of the lower level programming problem are extended and a sufficient condition for CRCQ to hold for VS form of BLPP with linear lower level programming problem is also given.展开更多
完整的三维地震数据的频率切片经过Hankel张量预变换后具有低秩性,因此,随机缺失的三维地震数据可以通过张量降秩进行插值。张量的秩最小化是NP难问题,传统方法通过构建张量秩的凸松弛求解。但是,该方法忽略张量模展开矩阵奇异值的差异...完整的三维地震数据的频率切片经过Hankel张量预变换后具有低秩性,因此,随机缺失的三维地震数据可以通过张量降秩进行插值。张量的秩最小化是NP难问题,传统方法通过构建张量秩的凸松弛求解。但是,该方法忽略张量模展开矩阵奇异值的差异性,获得的仅是原始张量秩最小化问题的次优解,地震数据插值精度得不到保证。为此,本文提出基于log_(ε)函数的非凸张量模型,用log_(ε)函数代替秩函数,采用交替方向乘子法(Alternating Direction Method of Multipliers,ADMM)求解。仿真和真实地震数据的实验结果表明,相比于传统的基于块Hankel变换的多道奇异谱分析(Multichannel Singular Spectrum Analysis,MSSA)方法和基于凸松弛的低秩张量补全方法(Low-rank Tensor Completion,LRTC),本文所提方法具有更高的重建精度。展开更多
基金This work was supported by the National Natural Science Foundation of China,the Oversea ExchangeFund of Nanjing Normal University,and CNPq of Brazil
文摘In this paper we present some algorithms for minimization of DC function (difference of two convex functions). They are descent methods of the proximal-type which use the convex properties of the two convex functions separately. We also consider an approximate proximal point algorithm. Some properties of the ε-subdifferential and the ε-directional derivative are discussed. The convergence properties of the algorithms are established in both exact and approximate forms. Finally, we give some applications to the concave programming and maximum eigenvalue problems.
基金Project supported by the National Natural Science Foundation of China(No.10071063)
文摘This paper presents a geometric characterization of convex sets in locally convex spaces onwhich a strong optimization theorem of the Stegall-type holds, and gives Collier's theorem ofw* Asplund spaces a localized setting.
基金supported by the National Natural Science Foundation of China(NSFC)(No.12001144)Zhejiang Provincial Natural Science Foundation of China(No.LQ20A010007)+1 种基金Chern Institute of Mathematicssupported by the National Natural Science Foundation of China(NSFC)(Nos.11871035,11531013).
文摘Two-phase image segmentation is a fundamental task to partition an image into foreground and background.In this paper,two types of nonconvex and nonsmooth regularization models are proposed for basic two-phase segmentation.They extend the convex regularization on the characteristic function on the image domain to the nonconvex case,which are able to better obtain piecewise constant regions with neat boundaries.By analyzing the proposed non-Lipschitz model,we combine the proximal alternating minimization framework with support shrinkage and linearization strategies to design our algorithm.This leads to two alternating strongly convex subproblems which can be easily solved.Similarly,we present an algorithm without support shrinkage operation for the nonconvex Lipschitz case.Using the Kurdyka-Lojasiewicz property of the objective function,we prove that the limit point of the generated sequence is a critical point of the original nonconvex nonsmooth problem.Numerical experiments and comparisons illustrate the effectiveness of our method in two-phase image segmentation.
文摘In this paper, we derive an exact penalty function for nonconvex bilevel programming problem based on its KS form. Based on this exact penalty function a sufficient condition for KS to be partially calm is presented and a necessary optimality condition for nonconvex bilevel programming problems is given. Some existing results about the differentiability of the value function of the lower level programming problem are extended and a sufficient condition for CRCQ to hold for VS form of BLPP with linear lower level programming problem is also given.
文摘完整的三维地震数据的频率切片经过Hankel张量预变换后具有低秩性,因此,随机缺失的三维地震数据可以通过张量降秩进行插值。张量的秩最小化是NP难问题,传统方法通过构建张量秩的凸松弛求解。但是,该方法忽略张量模展开矩阵奇异值的差异性,获得的仅是原始张量秩最小化问题的次优解,地震数据插值精度得不到保证。为此,本文提出基于log_(ε)函数的非凸张量模型,用log_(ε)函数代替秩函数,采用交替方向乘子法(Alternating Direction Method of Multipliers,ADMM)求解。仿真和真实地震数据的实验结果表明,相比于传统的基于块Hankel变换的多道奇异谱分析(Multichannel Singular Spectrum Analysis,MSSA)方法和基于凸松弛的低秩张量补全方法(Low-rank Tensor Completion,LRTC),本文所提方法具有更高的重建精度。
