本文選題：非凸非光滑優(yōu)化 + lower-C~2�。� 參考：《廣西大學(xué)》2017年碩士論文

【摘要】：本學(xué)位論文研究非光滑優(yōu)化(不可微優(yōu)化)問題,并且目標函數(shù)不一定是凸函數(shù),許多實際問題可以歸結(jié)為此類問題.因此,研究穩(wěn)定、高效的數(shù)值優(yōu)化方法求解非凸非光滑優(yōu)化問題有著重要的理論意義和實際價值.本文基于鄰近束方法和擬牛頓方法的思想,并結(jié)合局部凸化技術(shù)和Armi-jo 線搜索規(guī)則,提出求解非凸非光滑優(yōu)化的擬牛頓型束方法.在每次迭代,算法通過適當(dāng)?shù)牟呗愿戮植客够瘏?shù)ηe,不僅有效克服由非凸目標函數(shù)導(dǎo)致的線性化誤差可能是負數(shù)的情況,并且確保滿足下降性條件的候選點是目標函數(shù)在當(dāng)前鄰近中心處的近似鄰近點.進一步地,基于近似鄰近點構(gòu)造近似次梯度和近似擬牛頓方向作為線搜索方向.然后,通過判斷近似次梯度的范數(shù)是否減小決定步長的選取,要么取單位步長,要么借助Armijo線搜索規(guī)則計算步長.在溫和的假設(shè)下,證明了算法的全局收斂性,并討論了算法的收斂速度(線性收斂,超線性收斂).在最后,為驗證算法的有效性和穩(wěn)定性,本文借助數(shù)學(xué)軟件MATLAB進行編程,初步的數(shù)值實驗結(jié)果表明本文所提出的算法是有效的和穩(wěn)定性.
[Abstract]:In this dissertation, we study nonsmooth optimization (non-differentiable optimization) problems, and the objective function is not necessarily convex function, many practical problems can be attributed to this kind of problems. Therefore, it is of great theoretical significance and practical value to study the stable and efficient numerical optimization method for solving non-convex non-smooth optimization problems. Based on the idea of proximity beam method and quasi-Newton method, combined with local convexity technique and Armi-jo line search rule, a quasi-Newtonian beam method for solving nonconvex nonsmooth optimization is proposed in this paper. In each iteration, the local convexation parameter 畏 _ e is updated by the appropriate strategy, which not only effectively overcomes the case that the linearization error caused by the non-convex objective function may be negative. And it is ensured that the candidate point satisfying the descent condition is the approximate adjacent point of the objective function at the current adjacent center. Furthermore, approximate subgradient and approximate quasi-Newton direction are constructed based on approximate adjacent points as line search directions. Then, by judging whether the norm of the approximate subgradient decreases to determine the selection of step size, either the unit step size or the Armijo line search rule is used to calculate the step size. Under mild assumptions, the global convergence of the algorithm is proved, and the convergence rate (linear convergence, superlinear convergence) of the algorithm is discussed. Finally, in order to verify the validity and stability of the algorithm, this paper uses the mathematical software MATLAB to program. The preliminary numerical results show that the proposed algorithm is effective and stable.
【學(xué)位授予單位】：廣西大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O224

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本文編號：1867214