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Improving fault tolerance of a neural network has been studied for more than two decades. Various training algorithms have been proposed in sequel. The on-line node fault injection-based algorithm is one of these algorithms, in which hidden nodes randomly output zeros during training. While the idea is simple, theoretical analyses on this algorithm are far from complete. This paper presents its objective function and the convergence proof. We consider three cases for multilayer perceptrons (MLPs). They are: (1) MLPs with single linear output node; (2) MLPs with multiple linear output nodes; and (3) MLPs with single sigmoid output node. For the convergence proof, we show that the algorithm converges with probability one. For the objective function, we show that the corresponding objective functions of cases (1) and (2) are of the same form. They both consist of a mean square errors term, a regularizer term, and a weight decay term. For case (3), the objective function is slight different from that of cases (1) and (2). With the objective functions derived, we can compare the similarities and differences among various algorithms and various cases.