  Projected Gradient Descent denotes a class of iterative methods for solving
optimization programs. Its applicability to convex optimization programs has
gained significant popularity for its intuitive implementation that involves
only simple algebraic operations. In fact, if the projection onto the feasible
set is easy to compute, then the method has low complexity. On the other hand,
when the problem is nonconvex, e.g. because of nonlinear equality constraints,
the projection becomes hard and thus impractical. In this paper, we propose a
projected gradient method for Nonlinear Programs (NLPs) that only requires
projections onto the linearization of the nonlinear constraints around the
current iterate, similarly to Sequential Quadratic Programming (SQP). Although
the projection is easier to compute, it makes the intermediate steps unfeasible
for the original problem. As a result, the gradient method does not fall either
into the projected gradient descent approaches, because the projection is not
performed onto the original nonlinear manifold, or into the standard SQP, since
second-order information is not used. For nonlinear smooth optimization
problems, we analyze the similarities of the proposed method with SQP and
assess its local and global convergence to a Karush-Kuhn-Tucker (KKT) point of
the original problem. Further, we show that nonlinear Model Predictive Control
(MPC) is a promising application of the proposed method, due to the sparsity of
the resulting optimization problem. We illustrate the computational efficiency
of the proposed method in a numerical example with box constraints on the
control input and a quadratic terminal constraint on the state variable.
