求解高维非线性优化的并行分块对角拟牛顿法

Parallel Block-diagonal Quasi-Newton Method for High dimensional nonlinear optimization

在拟牛顿法的基础上提出了一种并行分块对角拟牛顿法.该方法在当前迭代点处用一个分块对角阵作为Hesse阵逆的近似,并在多个不同处理器中利用拟牛顿校正公式同时并行求解各个子矩阵,进而构造各个子方向.将各个子方向进行组合得到当前迭代点处的搜索方向,再利用并行Armijo线性搜索策略,将求解函数值的任务分配给多个不同处理器同时并行执行,求得搜索步长,从而求得下一个迭代点,直到收敛.数值算例结果表明该方法对高维非线性无约束优化问题具有良好的收敛性,并在保证计算精度的同时,显著地提高了计算效率,减少了计算时间.

A parallel block - diagonal quasi - Newton method is proposed based on quasi - Newton method. This method adopts a block - diagonal matrix as an approximation of the reverse of Hessian matrix at current iteration point, and then every block sub - matrix is computed through quasi - Newton update formula on several different processors at the same time, and then every sub - direction is computed. These sub - directions are assembled to get the search direction at current iteration point, and then parallel Armijo linear search strategy is used to achieve search step and then next iteration point, with each processor solving a function value at the same time. This process is done repeatedly until convergence. The results of numerical examples show that this method has good convergence for high dimensional nonlinear unconstraint optimization, and it can improve computing efficiency and reduce computing time markedly, and can ensure enough computing precision at the same time.