Summary: We present a numerical method based on surrogate duality to solve contact problems in elasticity. The primal optimization problem arising from the contact problem is converted to a corresponding dual problem using surrogate duality. An explicit dual formulation for the surrogate dual problem is obtained under certain conditions, and a Karmarkar’s interior point based method is introduced to solve the explicit surrogate dual problem. Numerical examples including some quadratic programming problems with dense matrices and one elastic contact problem are solved using the present method to show its feasibility and efficiency.