The Algorithm of Sweep-the-Negatives and its Applications to Order Resticted Inference

Consider the extreme problem : min$(\mu-y)'(\mu-y)$ subject to $A\mu \geq 0$, where A is an $n \times p$ matrix, which often occurs in solving the maximum likelihood estimator with ordered restrictions in parameter space. In case the matrix AA' has all non-positive off-diagonal elements, some propositions in this paper guarantee that the above extreme solutions are achieved at most at n sweep out steps in Gaussian eliminations. Some typicla examples of Sweep-the-Negatives method are given.