(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.
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.