  In equality-constrained optimization, a standard regularity assumption is
often associated with feasible point methods, namely the gradients of
constraints are linearly independent. In practice, the regularity assumption
may be violated. To avoid such a singularity, we propose a new projection
matrix, based on which a feasible point method for the continuous-time,
equality-constrained optimization problem is developed. First, the equality
constraint is transformed into a continuous-time dynamical system with
solutions that always satisfy the equality constraint. Then, the singularity is
explained in detail and a new projection matrix is proposed to avoid
singularity. An update (or say a controller) is subsequently designed to
decrease the objective function along the solutions of the transformed system.
The invariance principle is applied to analyze the behavior of the solution. We
also propose a modified approach for addressing cases in which solutions do not
satisfy the equality constraint. Finally, the proposed optimization approaches
are applied to two examples to demonstrate its effectiveness.
