求解非凸非光滑优化的拟牛顿型束方法

发布时间：2018-05-09 19:58

本文选题：非凸非光滑优化 + lower-C~2　；参考：《广西大学》2017年硕士论文

【摘要】：本学位论文研究非光滑优化(不可微优化)问题,并且目标函数不一定是凸函数,许多实际问题可以归结为此类问题.因此,研究稳定、高效的数值优化方法求解非凸非光滑优化问题有着重要的理论意义和实际价值.本文基于邻近束方法和拟牛顿方法的思想,并结合局部凸化技术和Armi-jo 线搜索规则,提出求解非凸非光滑优化的拟牛顿型束方法.在每次迭代,算法通过适当的策略更新局部凸化参数ηe,不仅有效克服由非凸目标函数导致的线性化误差可能是负数的情况,并且确保满足下降性条件的候选点是目标函数在当前邻近中心处的近似邻近点.进一步地,基于近似邻近点构造近似次梯度和近似拟牛顿方向作为线搜索方向.然后,通过判断近似次梯度的范数是否减小决定步长的选取,要么取单位步长,要么借助Armijo线搜索规则计算步长.在温和的假设下,证明了算法的全局收敛性,并讨论了算法的收敛速度(线性收敛,超线性收敛).在最后,为验证算法的有效性和稳定性,本文借助数学软件MATLAB进行编程,初步的数值实验结果表明本文所提出的算法是有效的和稳定性.
[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.
【学位授予单位】：广西大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O224

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