求解一类非凸非光滑优化问题的邻近交替束方法

发布时间：2018-09-03 16:35

【摘要】：非光滑优化在图像去噪、神经网络学习、经济学及计算化学和物理学等领域广泛使用,按目标函数与约束函数的凸性可分为凸非光滑优化和非凸非光滑优化.本学位论文研究一类非凸非光滑无约束优化问题,其目标函数具有一种特殊结构,它是由一个非凸函数及一个凸函数构成的和函数.此类问题应用非常广泛,如图像的恢复处理、压缩感知、最优控制、系统识别等.因此,研究这类问题的求解具有理论意义和实用价值.本文提出了一种邻近交替束方法来求解由非凸函数和凸函数构成的和函数的非凸非光滑优化问题.首先,利用局部凸化技术对非凸函数进行局部凸化处理,即将目标函数中的非凸函数项增加一个二次项;其次,构造局部凸化函数的割平面模型近似非凸函数项;最后,通过交替线性化方法分别对割平面模型和凸函数交替线性化得到两个简单的子问题.在算法的设计中,每一次迭代仅需求解两个简单的子问题.此方法将传统的交替线性束方法由凸推广到非凸情形.此外,在算法中我们设计新型邻近参数和凸化参数的调整策略,以保证算法的全局收敛性.本文的最后,我们对所提出的算法进行数值试验,数值结果验证了算法的可行性、有效性及稳定性.
[Abstract]:Non-smooth optimization is widely used in image denoising, neural network learning, economics, computational chemistry and physics. According to the convexity of objective function and constraint function, it can be divided into convex non-smooth optimization and non-convex non-smooth optimization. In this paper, we study a class of nonconvex nonsmooth unconstrained optimization problems. The objective function has a special structure, which is composed of a nonconvex function and a convex function. Such problems are widely used, such as image restoration, compression perception, optimal control, system recognition and so on. Therefore, it is of theoretical significance and practical value to study the solution of this kind of problem. In this paper, a adjacent alternating beam method is proposed to solve the nonconvex nonsmooth optimization problem of sum functions composed of nonconvex functions and convex functions. Firstly, the local convexity of non-convex function is treated by using the local convexity technique, which adds a quadratic term to the non-convex function in the objective function. Secondly, the cut-plane model of the local convexity function is constructed to approximate the non-convex function term. Two simple subproblems are obtained by alternating linearization for cut plane model and alternating linearization of convex function. In the design of the algorithm, only two simple subproblems need to be solved for each iteration. This method extends the conventional alternating linear beam method from convex to non-convex. In addition, in order to ensure the global convergence of the algorithm, we design a new adjustment strategy for adjacent parameters and convexity parameters. At the end of this paper, numerical experiments are carried out to verify the feasibility, validity and stability of the proposed algorithm.
【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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