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郑权等在[1]-[3]中提出了一种求解无约束优化问题的均值算法,若假设目标函数f(x)是连续的,还讨论了均值算法的收敛性。若假设f(x) 有界闭集Ω上的凸函数,本文证明了求解凸函数极小值的均值算法是线性收敛的。 相似文献
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信赖域算法是求解无约束优化问题的一种有效的算法.对于该算法的子问题,本文将原来目标函数的二次模型扩展成四次张量模型,提出了一个带信赖域约束的四次张量模型优化问题的求解算法.该方法的最大特点是:不仅在张量模型的非稳定点可以得到下降方向及相应的迭代步长,而且在非局部极小值点的稳定点也可以得到下降方向及相应的迭代步长,从而在算法产生的迭代点列中存在一个子列收敛到信赖域子问题的局部极小值点. 相似文献
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本文研究了稀疏分裂可行问题.通过将分裂可行问题转化为一个目标函数为凸函数的稀疏约束优化问题,设计一种梯度投影算法来求解此问题,获得了算法产生的点列可以收敛到稀疏分裂可行问题的一个解.用数值例子说明了算法的有效性. 相似文献
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ADMM算法是求解可分离凸优化问题的经典算法之一,但其无法保证原始迭代序列的收敛性且其子问题计算量很大.为了保证该算法所有迭代点列的全局收敛性及提高计算效率,采用凸组合技术的黄金比率邻近ADMM算法被提出,其中凸组合因子Ψ是关键参数.本文在黄金比率邻近ADMM算法的基础上,扩大了凸组合因子Ψ的取值范围,提出了收敛步长范围更广的推广黄金比率邻近ADMM算法.并在一定的假设下,证明了算法的全局收敛性及函数值残差和约束违反度在遍历意义下的O(1/N)次线性收敛速度.以及,当目标函数中任意一个函数强凸时,证明了算法在遍历意义下的O(1/N2)收敛率.最后,本文通过数值试验表明推广算法的有效性. 相似文献
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交替方向乘子法(ADMM)是一种求解可分离优化问题的简单有效的方法,相关研究已经较为完善.然而,当目标函数存在耦合项时,对ADMM算法收敛性的研究还处于初期.文章针对非凸非光滑不可分离优化问题,基于对称交替方向乘子法(SADMM),结合线性化技术,提出了一种新的线性对称邻近ADMM.在一定的假设条件下,证明了算法生成的序列有界并收敛至增广拉格朗日函数的稳定点.其次,当辅助函数满足Kurdyka-Lojasiewicz性质时,证明了算法的强收敛性.最后,数值实验的结果表明了算法的有效性. 相似文献
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近年来,关于多个凸函数和的优化问题受到广泛关注.本文研究三个凸函数和f(x)+g(x)+h(Bx)的一类凸优化问题,其中f (x)可微且具有Lipschitz连续梯度, g(x)和h(x)是正则下半连续简单凸函数, B是一个有界线性算子.此类优化问题在信号恢复和图像处理等实际问题中有着广泛的应用.为充分利用问题中的可微函数,本文基于向前向后分裂算法和三算子分裂算法框架,建立若干具有内外迭代形式的算法.在推导迭代算法的过程中,本文提出基于对偶和原始对偶方法求解函数g+h?B和h?B的邻近算子.在对参数一定假设条件下,本文证明所提出的迭代算法收敛性.通过与Condat和Vu算法、原始对偶不动点(primal-dual?xed point, PDFP)算法和原始对偶三算子(primal-dual three-operator, PD3O)算法比较,建立三种迭代算法与本文提出的迭代算法之间的联系.最后,通过对融合Lasso问题、约束全变分正则化问题和低秩全变分图像超分辨率重建问题实施一系列数值实验,验证所提出的迭代算法的有效性. 相似文献
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G. E. Ivanov 《Mathematical Notes》2006,79(1-2):55-78
In this paper, the notion of a weakly convex set is introduced. Sharp estimates for the weak convexity constants of the sum and difference of such sets are given. It is proved that, in Hilbert space, the smoothness of a set is equivalent to the weak convexity of the set and its complement. Here, by definition, the smoothness of a set means that the field of unit outward normal vectors is defined on the boundary of the set; this vector field satisfies the Lipschitz condition. We obtain the minimax theorem for a class of problems with smooth Lebesgue sets of the goal function and strongly convex constraints. As an application of the results obtained, we prove the alternative theorem for program strategies in a linear differential quality game. 相似文献
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Banach空间上凸泛函的某些对偶性质 总被引:2,自引:0,他引:2
本语文对Banach空间上的凸泛函建立了若干对偶性质,通过对偶的手段,建立了非常简洁的凸泛函极小化序列弱收敛的特征定理。 相似文献
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Generalized polyhedral convex sets, generalized polyhedral convex functions on locally convex Hausdorff topological vector spaces, and the related constructions such as sum of sets, sum of functions, directional derivative, infimal convolution, normal cone, conjugate function, subdifferential are studied thoroughly in this paper. Among other things, we show how a generalized polyhedral convex set can be characterized through the finiteness of the number of its faces. In addition, it is proved that the infimal convolution of a generalized polyhedral convex function and a polyhedral convex function is a polyhedral convex function. The obtained results can be applied to scalar optimization problems described by generalized polyhedral convex sets and generalized polyhedral convex functions. 相似文献
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We consider the distance function (DF), given by the caliber (the Minkowski gauge function) of a convex body, from a point to strictly, strongly, and weakly convex sets in an arbitrary Hilbert space. Some properties of the caliber of a strongly convex set and the conditions for obtaining a strict, strong, or weak convexity of Lebesgue sets for the distance function are established in accordance with the requirements for the set, the caliber of which specifies the distance function, and the set to which the distance is measured. The corresponding inequalities are obtained that reflect the behavior of the distance function on segments and allow comparing it with strictly, strongly, or weakly convex functions. 相似文献
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Nonlinear Proximal Decomposition Method for Convex Programming 总被引:2,自引:0,他引:2
In this paper, we propose a new decomposition method for solving convex programming problems with separable structure. The proposed method is based on the decomposition method proposed by Chen and Teboulle and the nonlinear proximal point algorithm using the Bregman function. An advantage of the proposed method is that, by a suitable choice of the Bregman function, each subproblem becomes essentially the unconstrained minimization of a finite-valued convex function. Under appropriate assumptions, the method is globally convergent to a solution of the problem. 相似文献
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几何凸函数与琴生型不等式 总被引:20,自引:3,他引:17
吴善和 《数学的实践与认识》2004,34(2):155-163
给出几何凸函数的定义以及判定几何凸函数的方法 ,建立关于几何凸函数的琴生型不等式 ,最后给出它的应用 ,包括改进一些已知不等式和建立一些新不等式 . 相似文献
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引入了一个广义凸函数一高阶强Pre-invex函数,它是Guneer-Bhatia介绍的高阶强凸函数的一种推广.在此基础上,讨论并证明了高阶强Pre-invex函数的一些等价刻画,这些结论是Guneer-Bhatia给出结论的推广,扩大了凸函数的应用范围. 相似文献
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称局部凸空间(E,(?)0)为WCM空间若对于任何弱于(?)0的局部凸拓扑(?),(E,(?))与(E,(?)0)具相同的弱紧圆凸集.本文研究了WCM空间的存在性及其与其他类型局部凸空间之间的关系,还给出了WCM空间的一种映照特征. 相似文献
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Global Optimality Conditions in Maximizing a Convex Quadratic Function under Convex Quadratic Constraints 总被引:2,自引:0,他引:2
Jean-Baptiste Hiriart-Urruty 《Journal of Global Optimization》2001,21(4):443-453
For the problem of maximizing a convex quadratic function under convex quadratic constraints, we derive conditions characterizing a globally optimal solution. The method consists in exploiting the global optimality conditions, expressed in terms of -subdifferentials of convex functions and -normal directions, to convex sets. By specializing the problem of maximizing a convex function over a convex set, we find explicit conditions for optimality. 相似文献
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