非凸优化问题的神经网络算法与应用的研究

发布时间：2018-03-11 11:27

本文选题：神经网络　切入点：伪凸　出处：《哈尔滨理工大学》2017年硕士论文　论文类型：学位论文

【摘要】：优化问题在科学和工程应用中随处可见。神经网络能获得优化问题的实时解,所以利用神经网络对优化问题求解已经得到了广泛的关注。本文针对伪凸优化问题与非凸二次优化问题两类情形,基于投影理论,极值映射以及法锥相关性质,分别构造离散和连续型神经网络的求解算法;并探讨了其在求解支持向量机中的应用。全文共分以下三个部分:针对伪凸优化问题,提出了一个离散型神经网络模型。首先,利用Karush-Kuhn-Tucker(KKT)条件和投影理论构造投影方程,使得投影方程的解与优化问题的解一一对应;进一步基于投影方程建立离散神经网络;理论结果表明,网络的平衡点与优化问题的最优解相对应,且网络具有全局指数收敛性。相比于连续网络,本文所构造的网络结构简单,减少了计算的复杂度;所得理论结果保证了网络能够有效求解伪凸优化问题。最后,利用数值算例进行仿真,数值结果表明了所设计的网络求解伪凸优化问题的有效性。针对非凸二次规划问题,首先根据规划问题的等式约束与不等式约束,分别构造了不同的罚函数。然后,利用极值映射以及正则函数的相关性质构造一微分包含形式的神经网络模型。针对所构造的网络模型,基于微分包含相关性质证明了罚函数沿着所构造的网络能在有限时间收敛至可行域内。其次,对网络的平衡点与临界点的一致性进行了证明,并基于切锥与法锥的性质分析了优化问题的最优点集与临界点集之间的关系。最后,在适当的假设条件下,对网络的收敛性进行了严格的证明。针对支持向量机的分类与回归问题导出的二次规划问题,利用所构造的离散神经网络进行求解。数值结果表明本文所构造的网络在求解支持向量机问题中表现出优良的性能。
[Abstract]:Optimization problems can be found everywhere in scientific and engineering applications. Neural networks can obtain real-time solutions of optimization problems. Therefore, the use of neural networks to solve optimization problems has been widely concerned. In this paper, for pseudoconvex optimization problems and non-convex quadratic optimization problems, based on projection theory, extremum mapping and normal cone correlation properties. The algorithms of discrete and continuous neural networks are constructed, and their applications in solving support vector machines are discussed. The paper is divided into three parts: a discrete neural network model is proposed for pseudoconvex optimization problems. The projection equation is constructed by using Karush-Kuhn-Tuckern KKT condition and projection theory. The solution of the projection equation corresponds to the solution of the optimization problem one by one. The discrete neural network is further established based on the projection equation. The equilibrium point of the network corresponds to the optimal solution of the optimization problem, and the network has global exponential convergence. Compared with the continuous network, the network structure in this paper is simple and the computational complexity is reduced. The theoretical results ensure that the network can effectively solve the pseudoconvex optimization problem. Finally, the numerical simulation results show that the designed network is effective in solving the pseudoconvex optimization problem. Firstly, different penalty functions are constructed according to equality constraints and inequality constraints of programming problems. A neural network model of differential inclusion is constructed by using extreme mapping and related properties of regular functions. Based on the correlation properties of differential inclusions, it is proved that the penalty function can converge to the feasible region in finite time along the constructed network. Secondly, the consistency between the equilibrium point and the critical point of the network is proved. Based on the properties of tangent cone and normal cone, the relationship between the optimal set and the critical point set of the optimization problem is analyzed. The convergence of the network is proved strictly. For the quadratic programming problem derived from the classification and regression problems of support vector machines, Numerical results show that the network presented in this paper has excellent performance in solving support vector machine problems.
【学位授予单位】：哈尔滨理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TP183;O221

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