A New Continuous-Time Equality-Constrained Optimization Method to Avoid Singularity

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. Index Terms Optimization, equality constraints, continuous-time dynamical systems, singularity