广义梯度系统与外插邻近算法的收敛性分析

发布时间：2018-01-13 18:28

本文关键词：广义梯度系统与外插邻近算法的收敛性分析　出处：《哈尔滨工业大学》2017年博士论文　论文类型：学位论文

【摘要】：本文首先研究了一类二阶梯度系统的收敛行为,基于该系统,进一步研究了几类外插邻近算法的收敛性与收敛速度。具体研究内容如下:1.研究了一类二阶梯度系统的收敛行为及其与外插邻近梯度算法之间的关系。首先,针对一类非凸解析势函数,利用?ojasiewicz不等式,在耗散项消失足够慢的条件下,证明了该系统的解轨道是收敛的,并且轨道长度有限。然后,讨论了二阶梯度系统与几类外插邻近梯度算法之间的关系。2.研究了一类外插邻近梯度算法的收敛行为,该算法用于求解一类非凸非光滑最小化问题。利用误差界条件,在外插项系数的上确界小于一个固定阈值的条件下,证明了由外插邻近梯度算法生成的迭代序列与函数值序列都是R线性收敛的。除此之外,当问题变成凸问题时,指出外插系数的阈值退化到1,进一步说明带有固定重启策略的快速迭代收缩阈值算法(fast iterative shrinkage-thresholding algorithm,简写为FISTA)是外插邻近梯度算法的一个特例。进而,利用带有固定重启策略的FISTA求解凸优化问题时,如果目标函数满足误差界条件,由该算法生成的迭代序列与函数值序列都是R线性收敛的。3.考虑了一类外插邻近梯度算法的收敛行为,该算法用于求解一类凸优化问题。对于一大类外插系数,包括FISTA中的外插系数,证明了由外插邻近梯度算法生成的迭代序列的连续变化趋于0.利用?ojasiewicz不等式,在外插系数满足一定条件下,证明了由外插邻近梯度算法生成的迭代序列是收敛的,并且序列长度有限。4.研究了外插邻近凸函数的差算法(difference-of-convex algorithm,简写为DCA)的收敛行为,该算法用于求解一类凸函数的差(difference-of-convex,简写为DC)优化问题。对于一大类外插系数,包括带有固定重启策略的FISTA中的外插系数,证明了由外插邻近DCA生成的迭代序列的任何一个聚点都是DC问题的一个平衡点。进一步,在目标函数满足一定条件下,利用Kurdyka-?ojasiewicz不等式,建立了外插迫近DCA的全局收敛性,并且分析了它的收敛速度。外插邻近DCA的有效性通过对带有DC正则函数的最小二乘问题做数值实验得以验证。
[Abstract]:This paper studies the convergence behavior of a class of two degree ladder system, based on the system, further study of the convergence and convergence rate of several adjacent extrapolation algorithm. The specific contents are as follows: 1. to investigate the relationship between the convergence behavior of a class of two degree system and step extrapolation between adjacent gradient algorithm. First of all, for a class of non convex analytic potential function, using the ojasiewicz inequality, disappear? Slow enough in terms of dissipation, proved that the system trajectory is convergent, and the track length is limited. Then, discuss the relationship between the two steps of.2. system and several kinds of interpolation between adjacent gradient algorithm on the convergence behavior a kind of extrapolation adjacent gradient algorithm, the algorithm for solving a class of nonconvex nonsmooth minimization problem. By using the error bound condition, inserted coefficient in the supremum is less than a fixed threshold conditions, proved by Iterative sequence and function generation gradient algorithm adjacent value sequence is R linear convergence. In addition, when the problem into a convex problem, pointed out that the extrapolation coefficient threshold degradation to 1, further explained the restart strategy with fixed fast iterative shrinkage thresholding algorithm (fast iterative shrinkage-thresholding algorithm, abbreviated as FISTA) is a a special case of extrapolation adjacent gradient algorithm. Then, using a fixed FISTA for solving convex optimization problem to restart strategy, if the target function satisfies the error bound condition, iterative sequence generated by the algorithm and function values of the sequences are R linear convergence.3. considering the convergence behavior of a class of extrapolation adjacent gradient algorithm, the the algorithm for solving a class of convex optimization problems. For a large class of interpolation coefficient, including FISTA extrapolation coefficient, it is proved that the extrapolation iterative gradient algorithm to generate neighboring sequences of continuous The change tends to 0. using? Ojasiewicz inequality, interpolation coefficient satisfies certain conditions in abroad, proved that the iterative sequence generated by extrapolation adjacent gradient algorithm is convergent, and the sequence length of.4. finite difference algorithm of extrapolation adjacent convex function (difference-of-convex algorithm, abbreviated as DCA) convergence behavior, the algorithm used to solve the A class of convex function difference (difference-of-convex, abbreviated as DC) optimization problem. For a large class of interpolation coefficient, including a fixed restart strategy in FISTA extrapolation coefficient, it is proved that any accumulation of the sequence generated by extrapolation near the DCA are a balance problem. Further DC the use of Kurdyka-, in which the objective function satisfies certain conditions? Ojasiewicz inequality, the global convergence is established and the impending DCA extrapolation, analyzes its convergence. The validity of the adjacent DCA by extrapolation The least square problem with DC canonical functions is verified by numerical experiments.

【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O224

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