一个求解约束非线性优化问题的微分方程方法

本文构造的求解非线性优化问题的微分方程方法包括两个微分方程系统,第一个系统基于问题函数的一阶信息,第二个系统基于二阶信息．这两个系统具有性质:非线性优化问题的局部最优解是它们的渐近稳定的平衡点,并且初始点是可行点时,解轨迹都落于可行域中．我们证明了两个微分方程系统的离散迭代格式的收敛性定理和基于第二个系统的离散迭代格式的局部二次收敛性质．还给出了基于两个系统的离散迭代方法的数值算例,数值结果表明基于二阶信息的微分方程方法速度更快．

A DIFFERENTIAL EQUATION METHOD FOR SOLVING NONLINEARLY CONSTRAINED OPTIMIZATION PROBLEMS

The differential equation method in this paper consists of two differential equation systems, in which the first one is based on the first order information on problem functions and the second system is based on the second order information. These two systems possess the properties that the local minimum point is their asymptotically stable equilibrium point and the whole solution trajectories are in the feasible region of the problem if they start from initial feasible points. We prove the convergence theorems for their discrete schemes and the locally quadratic convergence property for the discrete method based on the second differential equation system. We give numerical examples based on these two discrete methods and the numerical results show that the differential equation system based on the second information is faster than the first one.