不等式约束非线性规划的拉格朗日乘子的收敛

对于等式约束的非线性规划问题,一般的解决方法是在每次迭代中更新拉格朗日乘子且逐渐增大拉格朗日函数的惩罚因子,当罚因子充分大或充分接近局部最优解时,二阶充分条件是满足的;对不等式约束问题也采用了相应的方法.在凸的情况下,对于任意的罚因子或者在每次迭代中不要求精确极小化,就能全局收敛到最优解;证明了拉格朗日乘子是收敛的.

The Convergence of Lagrange Multiplier for Nonlinear Programming with General Inequality Constraints

As for nonlinear programming problem with equality constraints, a general method of solution is to gradually increase penalty factor of Lagrange function and to renew Lagrange multipliers in the iteration of each cycle. If penalty factor is sufficiently large or is close to local optimal solution, the second-order sufficient conditions are satisfied.This paper use the corresponding method for inequality-constrained problems. Global convergence to an optimal solution is established in the convex case for an arbitrary penalty factor and without the requirement of an exact minimum in the iteration of each cycle.Furthermore, the Lagrange multipliers are proved to converge.