This problem is NP-hard, so to get a good approximation solution we used convex semidefinite relaxation with different types of constraints. This approach is the powerful optimization technique that helps to reformulate the initial non-convex problem as a convex one and sometimes even gives the exact solution. The important step is to operate with precise energy function, which is used to compute the energy of different interactions in proteins. To obtain this, we used the machine-learning procedure, which extracts the parameter vector for the potential from the training set of protein structures. After the training step, we used this vector to compute the energy of a protein and to find the side-chains corresponding to the minimal total energy of the protein. The current accuracy in side-chain prediction is about $80\%$, which is achieved using the spectrum relaxation of the matrix in the quadratic term. Also, this approach is very fast, precisely, it requires less than 1 second per protein to predict the positions of its side-chains.