In dextrous robotic hand grasping the external object force needs to be balanced with contact forces maintaining grasp stability. Redundancy in the grasp yields the optimization problem to minimize the grasp effort subject to nonlinear friction stability constraints. We reformulate the optimization problem as a semidefinite program with affine constraints. In this paper, two versions of strictly convex cost functions including a barrier term, one of them self-concordant, are considered. For the general class of such cost functions Dikin-type algorithms are considered. It is shown that the proposed algorithms guarantee convergence to the unique solution of the underlying semidefinite program. Numerical examples demonstrate the simplicity of implementation and the good numerical properties of the approach.