We consider an heterogeneous elasticity problem and we show how to approximate it using a problem of the same type, but with effective constant coefficients that are defined, in an inverse problem spirit, by an optimization procedure. Our approach is based on the sole knowledge of the average displacement and stress at the boundary of the system, which are quantities that experimental measurements can provide. In the limit of infinitely small oscillations of the coefficients, we illustrate the links between this approach and the classical theory of homogenization. We also discuss comprehensive numerical tests and comparisons that show the practical interest of the approach. This is joint work with Claude Le Bris and Simon Ruget (Ecole des Ponts and Inria).