计算机科学 ›› 2019, Vol. 46 ›› Issue (11): 228-234.doi: 10.11896/jsjkx.181001926
喻昕, 马崇, 胡悦, 伍灵贞, 汪炎林
YU Xin, MA Chong, HU Yue, WU Ling-zhen, WANG Yan-lin
摘要: 优化问题的研究一直以来深受科研工作者的关注,凸优化问题作为优化问题的一个重要部分更是成为研究重点,许多应用神经网络思想提出的模型已经被应用到实际问题中。然而,在机器学习、信号处理、生物信息学等领域中涉及的优化问题往往不是凸优化问题,而是伪凸优化及非凸优化的问题,因此解决后一类问题变得刻不容缓。针对目标函数是非光滑伪凸函数、约束函数由等式和不等式函数构成的优化问题,基于罚函数以及微分包含的思想,构建了一个新型的不含惩罚参数的单层神经网络模型。该模型的主要设计思路是根据已经提出的神经网络模型思想,为目标函数的梯度设计一个制约的函数,使其值始终保持在一个范围之内,再结合一个关于时间的函数,确保其值随时间变小。同时,考虑到不等式约束对状态解进入等式约束之前的收敛方向有影响,加入一个条件函数来限制它。与已提出的神经网络模型相比,所提模型具有结构简单、无须提前进行惩罚参数的计算、对初始点的位置无特殊要求等优势。而且,对于任意初始点,理论证明了状态解的有界性、状态解能够在有限时间内收敛到等式约束内并永驻其中、状态解能够在有限时间内收敛到可行域并永驻其中以及状态解最终收敛到优化问题的最优解。在MATLAB环境下,通过数学仿真实验,状态解都能快速地收敛到一个最优解。同时,用已经提出的类似神经网络模型解决同样的优化问题时,若罚参数或初始点选择不恰当则会导致状态解不能很好地收敛。这不仅验证了所提出的理论结果的正确性,同时也说明了所提网络具有更广泛的应用范围。
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