In this paper, a criterion is proposed in the form of a quadratic function for the purpose of self-motion planning of redundant robot arms. The proposed self-motion scheme with joint physical limits considered could be formulated as a quadratic programming (QP) problem subject to equality, (inequality) and bound constraints. A primal-dual neural network based on linear variational inequalities (LVI) is developed as the real-time solver for the resultant quadratic-program. The so-called LVI-based primal-dual neural network has a simple piecewise-linear dynamics and a global exponential convergence to optimal solutions of QP problems. Computer-simulations performed based on PA10 robot arm substantiate the efficacy of the proposed QP-based neural self-motion-planning scheme.