基于对角隐式Runge-Kutta公式的无约束优化方法

Methods based on semi-diagonal implicit Runge-Kutta formulae for unconstrained optimization

通过把一个无约束优化问题转化为一个等价的常微分方程,利用二阶半对角隐式Runge Kutta公式构造了求解无约束优化问题的LRKOPT算法。LRKOPT算法具有与IMPBOT方法相似的数值特性,但LRKOPT算法可以看成是最速下降方向与牛顿法方向的非线性组合,而IMPBOT方法为它们两者之间的线性组合。在目标函数为一致凸函数的假设条件下,证明了LRKOPT方法的具有全局收敛和局部超线性收敛性。数值结果表明LRKOPT方法具有很好的数值稳定性并且LRKOPT方法的计算效率优于IMPBOT方法。

A LRKOPT algorithm of solving unconstrained optimization problem is constructed using two-orders semi-diagonal implicit Runge-Kutta formulas through transforming the unconstrained optimization problem to the equivalant ordinary differential equtions. The LRKOPT algorithm possesses numerical performance being simillar to IMPBOT method, but the LRKOPT algorithm can regard as a nonlinear composition between the steepest descent direction and the Newton direction and the IMPBOT method as its linear composition. Under the assumption condition of taking target function as an uniform convex function. We have proved that the LRKOPT has the global convergence and partial superlinear convergence. Numerical results show that the LRKOPT method has better numerical stability and its calculation efficiency prior to the IMPBOT one.