There is considered an image recognition problem, defined for the single hidden layer perceptron, fed with 5-by-7 monochrome images on its input under Gaussian noise of their distortion. In this neural network the hidden layer neuron number should be set optimally to maximize its productivity. For minimizing traintime duration and recognition error rate both simultaneously there are suggested two ways of solving the corresponding two-objective minimization problem. One of them deals with equilibrium conception, and the other takes Bernoulli criterion for getting the single minimization problem.