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Kinematic redundancy of robot will lead to the existence of an infinite number of inverse kinematics solutions which makes it possible to select a "best" solution according to an optimization criterion. In this paper, two optimization objective functions are proposed aiming at either minimizing extra degrees of freedom (DOFs) or minimizing potential energy of a multi-link redundant robot. Physical constraints, either equality or inequality types, are taken into consideration in the objective functions. Since the closed-form solutions do not exist in general for highly nonlinear and constrained optimization problems, we selected and developed two numerical methods which are verified to be effective and precise in solving the two optimization problems associated with the redundant inverse kinematics. We first verify that the well established trajectory following method can precisely solve the two optimization problems, but takes certain amount of computation time. To speed up, sequential method that combines the sequential quadratic programming and iterative Newton-Raphson algorithm is developed. A 4 DOFs FUJITSU HOAP-1 humanoid robot arm is used as the prototype to validate the effectiveness of the proposed optimization solutions.