(2) The convexification method for the collision avoidance constraints and the sequential

convex programming for solving the underlying optimization problem in MPC are effective.

The proposed method is based on minimizing a novel objective function in the form of a difference of

convex programming (DCP) problem.

Therefore, the constrained quadratic programming problem (46) is a

convex programming problem, by Theorem 20.

The numerical implementation of the two proposed approaches takes advantage of the CVX Matlab[R] toolbox [44, 45], a general software for

convex programming.

A Bisection Method for

Convex Programming Problem Solution based on Projection Neural Network.

As we know,

convex programming is a special kind of non-linear programming.

Thus, one can convert (MP) into the following equivalent reverse

convex programming problem (RCP):

We emphasize the theories of Nesterov and Nemirovsky [18] which use this class of functions for developing linear and convex quadratic programs with convex quadratic constraints, and Udriste's works [26], [28] which develop barrier methods for smooth

convex programming on Riemannian manifolds.

As a consequence, by virtue of the choices which have been made in choosing the cost functional, one can assume that also the cost functional is a convex one, so that the overall problem is reduced to a

Convex Programming (CP) one.

ADMM can be viewed as an application of the Douglas-Rachford splitting method to the dual of the two-block separable convex programming(2) [17] or a special case of the proximal point method for the general

convex programming [18], or a generalization of the classical Uzawa method for solving the saddle-point problems [19].

The concepts of the [phi]-convex and [phi]-invex functions have played very important role in the development of generalized

convex programming. From definitions 2.2 and 2.3, it is clear that the differentiable [phi]-convex function are [phi]-invex functions.

Fukushima, "Application of the alternating direction method of multipliers to separable

convex programming problems," Computational Optimization and Applications, vol.