求解约束最优化问题KKT系统的BFGS方法

A BFGS Method for Solving KKT Systems of Constrained Optimization Problems

利用Fischer-Burmeister函数,将约束最优化问题KKT系统转化为等价的非光滑方程组,利用广义导数,给出一个求解该非光滑方程组的BFGS方法.其子问题是一个系数阵为正定对称阵的线性方程组.为保证全局收敛性,我们引进了一个适当的线性搜索,它使得效益函数近似下降.在适当的条件下,我们证明了算法是适定的,并具有全局收敛性和超线性收敛性.

In this paper,by using the FischerBurmeister function,we reformulate the KKT system of a contrained optimization problem into an equivalent nonsmooth equation.We then propose a BFGS method for solving this nonsmooth equation reformilation.The subproblems of the proposed method are systems of linear equations with symmetric and positive define coefficient matrices.To ensure the global convergence,we introduce a nonmonotne line search under which the generated iterates exhibit an approximate norm descent property.Under suitable conditions,we show that the proposed method is welldefined and globally and superlinearly convergent to a KKT point of the constrained optimization problem.