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  本文關(guān)鍵詞：廣義梯度系統(tǒng)與外插鄰近算法的收斂性分析　出處：《哈爾濱工業(yè)大學(xué)》2017年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 梯度系統(tǒng) 外插鄰近算法 收斂 非凸問(wèn)題 ?ojasiewicz不等式 誤差界條件

【摘要】：本文首先研究了一類二階梯度系統(tǒng)的收斂行為,基于該系統(tǒng),進(jìn)一步研究了幾類外插鄰近算法的收斂性與收斂速度。具體研究?jī)?nèi)容如下:1.研究了一類二階梯度系統(tǒng)的收斂行為及其與外插鄰近梯度算法之間的關(guān)系。首先,針對(duì)一類非凸解析勢(shì)函數(shù),利用?ojasiewicz不等式,在耗散項(xiàng)消失足夠慢的條件下,證明了該系統(tǒng)的解軌道是收斂的,并且軌道長(zhǎng)度有限。然后,討論了二階梯度系統(tǒng)與幾類外插鄰近梯度算法之間的關(guān)系。2.研究了一類外插鄰近梯度算法的收斂行為,該算法用于求解一類非凸非光滑最小化問(wèn)題。利用誤差界條件,在外插項(xiàng)系數(shù)的上確界小于一個(gè)固定閾值的條件下,證明了由外插鄰近梯度算法生成的迭代序列與函數(shù)值序列都是R線性收斂的。除此之外,當(dāng)問(wèn)題變成凸問(wèn)題時(shí),指出外插系數(shù)的閾值退化到1,進(jìn)一步說(shuō)明帶有固定重啟策略的快速迭代收縮閾值算法(fast iterative shrinkage-thresholding algorithm,簡(jiǎn)寫為FISTA)是外插鄰近梯度算法的一個(gè)特例。進(jìn)而,利用帶有固定重啟策略的FISTA求解凸優(yōu)化問(wèn)題時(shí),如果目標(biāo)函數(shù)滿足誤差界條件,由該算法生成的迭代序列與函數(shù)值序列都是R線性收斂的。3.考慮了一類外插鄰近梯度算法的收斂行為,該算法用于求解一類凸優(yōu)化問(wèn)題。對(duì)于一大類外插系數(shù),包括FISTA中的外插系數(shù),證明了由外插鄰近梯度算法生成的迭代序列的連續(xù)變化趨于0.利用?ojasiewicz不等式,在外插系數(shù)滿足一定條件下,證明了由外插鄰近梯度算法生成的迭代序列是收斂的,并且序列長(zhǎng)度有限。4.研究了外插鄰近凸函數(shù)的差算法(difference-of-convex algorithm,簡(jiǎn)寫為DCA)的收斂行為,該算法用于求解一類凸函數(shù)的差(difference-of-convex,簡(jiǎn)寫為DC)優(yōu)化問(wèn)題。對(duì)于一大類外插系數(shù),包括帶有固定重啟策略的FISTA中的外插系數(shù),證明了由外插鄰近DCA生成的迭代序列的任何一個(gè)聚點(diǎn)都是DC問(wèn)題的一個(gè)平衡點(diǎn)。進(jìn)一步,在目標(biāo)函數(shù)滿足一定條件下,利用Kurdyka-?ojasiewicz不等式,建立了外插迫近DCA的全局收斂性,并且分析了它的收斂速度。外插鄰近DCA的有效性通過(guò)對(duì)帶有DC正則函數(shù)的最小二乘問(wèn)題做數(shù)值實(shí)驗(yàn)得以驗(yàn)證。
[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.

【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O224

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