In the first part of this talk, we introduce some tools that are often used in optimization theory (such as Γ-convergence and relaxation) to deal with minimization problems. In the second part, we present a shape optimization problem arising from the optimal reinforcement of a membrane using one-dimensional stiffeners. In particular, we consider the maximization of the first eigenvalue on several admissible classes and discuss the corresponding relaxed problems, existence results and open problems.