等式约束对病态问题的影响及约束正则化方法

有效利用参数间已知的等式约束信息能够提高最小二乘解的精度,消除秩亏,但是等式约束能否消除或减弱平差模型的病态性尚不明了,由此提出了一种通过消除部分参数将等式约束病态问题转化为无约束问题的方法。然后分析了等式约束对病态问题的影响,用简单实例证明了加入约束后,系统可能呈现良态或病态,它的性态由原设计阵和等式约束共同决定,并提出了求解等式约束病态问题的诊断-正则化两步方法。最后用一个数值实例验证了该方法的可行性。

Influence of Equality Constraints on Ill-conditioned Problems and Constrained Regularization Method

The accuracy of least square estimation can be improved and the rank deficient problem can be eliminated by using equality constraints properly. However, the influence of equality constraints in an ill-conditioned problem is not clear. Therefore, a method to transform the equality-constrained ill-posed to unconstrained problem by eliminating a part of the parameters is proposed in this paper. We firstly analyzed the influence of equality constraints on an ill-posed problem. Subsequently, the new system with equality constraints was verified by means of an example as either ill-conditioned or not depending on the original design matrix and equality constraints. We finally propose a diagnosis-regularization two-step approach to solve the equality-constrained ill-posed problem and validate its feasibility with a simulated data experiment.