求解非线性最优化问题的序列线性方程组算法

Sequential Systems of Linear Equations Algorithm for the Solution of Nonlinear Optimization Problems

序列二次规划(SQP)算法是目前公认的求解非线性约束优化问题的最有效的算法之一。但是目前SQP算法存在两个重要问题:(1)每步需要求解一至两个二次规划子问题以得到迭代方向,计算工作量大,难以应用于大规模问题;(2)迭代过程中产生的二次规划子问题可能无解,使运算过程中断。尽管可用其他措施重新定义迭代方向,但必然增加算法的复杂性,增大计算工作量,理论证明也不完善。文中介绍的序列线性方程组方法就是针对SQP算法的缺点而提出的。理论分析和数值实验均表明,这种算法具有迭代时间少,收敛速度快等优点,可以用来求解大规模的非线性优化问题。

Sequential Quadratic Programming （SQP） Algorithm is one of the most effective methods for solving nonlinear constrained optimization problems. However, there are still two main defults existing in current SQP algorithm. （1） At each iteration, an SQP algorithm need to solve one or two quadratic programs to obtain the search direction, which is much time-consuming and not fit for large-scale problems; （2） The quadratic subprogram generated during the iteration may have no solution, which will break out the procedure; Although we can redefine the search direction using other strategy, it is bound to increase the complexity of the algorithm and the computation amount; Moreover, the proof of convergence is not complete either. The Sequential Systems of Linear Equations （SSLE） method introduced in this paper is mainly to overcome the above two defaults of SQP method. Theoretical analysis and numerical experiments show that this kind of method is time-saving and has rapid convergence rate, which is much fit for the solution of large-scale optimization problems.