The material identification problem addressed consists of determining the constitutive parameters distribution of a linear elastic solid using displacement measurements. This problem has been considered in important applications such as the design of methodologies for breast cancer diagnosis. Since the resolution of real life problems involves high computational costs, there is great interest in the development of efficient methods. In this paper two new efficient formulations of the problem are presented. The first formulation leads to a second-order cone optimization problem, and the second one leads to a quadratic optimization problem, both allowing the resolution of the problem with high efficiency and precision. Numerical examples are solved using synthetic input data with error. A regularization technique is applied using the Morozov criterion along with an automatic selection strategy of the regularization parameter. The proposed formulations present great advantages in terms of efficiency, when compared to other formulations that require the application of general nonlinear optimization algorithms.