Motivated by the impressive success of deep learning in a wide range of scientific and industrial applications, we explore in this work the application of deep learning into a specific class of optimization problems lacking explicit formulas for both objective function and constraints. Such optimization problems exist in many design problems, e.g., rotor profile design, in which objective and constraint values are available only through experiment or simulation. They are especially challenging when design parameters are high-dimensional due to the curse of dimensionality. In this work, we propose a data-informed deep optimization (DiDo) approach emphasizing on the adaptive fitting of the the feasible region as follows. First, we propose a deep neural network (DNN) based adaptive fitting approach to learn an accurate DNN classifier of the feasible region. Second, we use the DNN classifier to efficiently sample feasible points and train a DNN surrogate of the objective function. Finally, we find optimal points of the DNN surrogate optimization problem by gradient descent. To demonstrate the effectiveness of our DiDo approach, we consider a practical design case in industry, in which our approach yields good solutions using limited size of training data. We further use a 100-dimension toy example to show the effectiveness of our approach for higher dimensional problems. Our results indicate that, by properly dealing with the difficulty in fitting the feasible region, a DNN-based method like our DiDo approach is flexible and promising for solving high-dimensional design problems with implicit objective and constraints.