本文選題：變分不等式 + LQP算法��； 參考：《重慶大學(xué)》2016年碩士論文

【摘要】：變分不等式有著廣泛的應(yīng)用背景,它是最優(yōu)化領(lǐng)域一類非常重要的研究工具。圖像恢復(fù)、信號(hào)處理、管理科學(xué)、統(tǒng)計(jì)計(jì)算、矩陣完整化、機(jī)器學(xué)習(xí)等信息技術(shù)領(lǐng)域中存在的大量凸優(yōu)化問題,均可以轉(zhuǎn)換成變分不等式問題來求解。在變分不等式這一統(tǒng)一框架下研究凸優(yōu)化問題的求解方法,常常會(huì)帶來很大的方便�？煞蛛x結(jié)構(gòu)型變分不等式是變分不等式的特殊形式,而LQP交替方向法是求解該類變分不等式的有效算法,該方法是原始交替方向法的一種改進(jìn)算法,它不僅充分利用原問題的可分離性,將原問題分裂為兩個(gè)較低維子問題,而且通過引入LQP正則項(xiàng),將子問題轉(zhuǎn)化為兩個(gè)更容易求解的非線性方程組,打破了原始交替方向法中必須求解兩個(gè)單調(diào)變分子問題的瓶頸。本文對(duì)LQP交替方向法及其改進(jìn)算法做了進(jìn)一步的探索。主要成果有以下兩個(gè)方面:(1)構(gòu)造一個(gè)新的下降方向,從而提出一種新的下降型LQP交替方向法。在算法分析的過程中給出了最優(yōu)步長的選取方式,并在較弱的假設(shè)條件下證明了算法全局收斂性。最后數(shù)值實(shí)驗(yàn)結(jié)果顯示新算法是可行的。(2)結(jié)合廣義交替方向法提出了一種非精確型LQP廣義交替方向法,其迭代格式只需要求解兩個(gè)子問題的近似解而非精確解。在選取適當(dāng)非精確準(zhǔn)則的條件下證明了算法的全局收斂性,最后給出了新算法在遍歷意義下和非遍歷意義下O(1/t)的收斂率分析,從而說明算法的有效性。
[Abstract]:Variational inequality has a wide application background, it is a very important research tool in the field of optimization. A large number of convex optimization problems in information technology such as image recovery, signal processing, management science, statistical computation, matrix integrity, machine learning and so on, can be transformed into variational inequality problems. It is very convenient to study the solution of convex optimization problems under the unified framework of variational inequalities. The separable structural variational inequality is a special form of variational inequality, and the LQP alternating direction method is an effective algorithm for solving this kind of variational inequality. This method is an improved algorithm of the original alternative direction method. It not only makes full use of the separability of the original problem and splits the original problem into two lower dimensional subproblems, but also transforms the subproblem into two more easily solved nonlinear equations by introducing the LQP regular term. It breaks the bottleneck of two monotonic molecular problems which must be solved in the original alternating direction method. In this paper, the LQP alternating direction method and its improved algorithm are further explored. The main results are as follows: 1) A new descent direction is constructed, and a new descending LQP alternating direction method is proposed. In the process of algorithm analysis, the selection of optimal step size is given, and the global convergence of the algorithm is proved under the condition of weak assumption. Finally, the numerical results show that the new algorithm is feasible. (2) an inexact LQP generalized alternating direction method is proposed in combination with the generalized alternating direction method. The iterative scheme only needs to solve the approximate solution of the two sub-problems rather than the exact solution. The global convergence of the algorithm is proved under the condition of selecting appropriate inexact criteria. Finally, the convergence rate analysis of the new algorithm in the sense of ergodic and non-ergodic is given, which shows the validity of the algorithm.
【學(xué)位授予單位】：重慶大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2016
【分類號(hào)】：O178

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 何炳生,蔣建林,錢邁建,許婭;PPA BASED PREDICTION-CORRECTION METHODS FOR MONOTONE VARIATIONAL INEQUALITIES[J];Numerical Mathematics A Journal of Chinese Universities(English Series);2005年01期

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本文編號(hào)：1826061