基于动力系统的无约束优化问题的方法分析

Analysis of Methods for Unconstrained Optimization Based on Dynamical Systems

通过解由常微分方程构成的动力系统的稳定点得到等价的无约束优化问题的局部极小点 ,而动力系统的稳定点可以沿动力系统轨线上的任一点通过路径跟踪得到。我们发现 ,在用Euler方法求解二次优化问题的等价动力系统的方程时 ,由方法的步长确定的稳定区域对应于这些方法所得到的迭代公式的步长满足单调下降算法的条件确定的单调下降区域 ,因此我们可以利用这个性质构造解无约束优化问题的数值方法而不采用标准的常微分方程的数值求解公式。分析了一些基于微分方程的无约束优化方法并举例说明这些方法有些是数值不可行的。

In this paper, we construct the differential equation to solve the unconstrained optimization problem. Such methods solving the ill-conditioned problems are very effective. It indicates that the asymptotically stable region of explicit the Euler method to solve the differential equation equals to the descent region of the gradient method to solve the original unconstrained optimization problem. We analyze the other numerical methods for unconstrained optimization problem based on dynamical systems and indicate that some methods may be not feasible.