We present a mathematical formulation to construct fractional factorial designs automatically. In experiments whose designs have many factors, it is sometimes difficult to try all possible combinations of the levels of the factors because of time and cost restriction. Therefore, for each experiment, we often use a fractional factorial design i.e. a design which consist of carefully chosen fraction of experimental runs of full factorial design. The difficulty here is that the criterion of the choice varies according to the situation. In this paper, we present a method to choose experimental runs from a full factorial desgin using the group lasso. The method allows us to obtain the optimal design matrix with respect to a given criterion by setting the tuning parameters adequately. We give some numerical examples to show that we can obtain the orthogonal arrays as the solutions of the group lasso problems
