We aim to efficiently determine a feasible region of an unknown black box function that assigns real numbers to discrete alternatives. Unlike existing binary classification methods that primarily focus on developing a classifier based on a given number of training data, we optimize the order of alternative sampling to achieve high accuracy in feasibility determination with a small number of observations. To achieve this, we utilize the Gaussian process as the surrogate model and introduce a novel value-of-information acquisition function to perform adaptive sampling under multiple constraints. We thoroughly analyze the convergence of our proposed scheme and demonstrate its effectiveness through numerical experiments.