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

TL;DR

This paper introduces a novel continuous-time optimization method that maintains feasibility and avoids singularities by using a new projection matrix, improving robustness in equality-constrained problems.

Contribution

A new projection matrix and continuous-time dynamical system are developed to handle singularities in equality-constrained optimization without regularity assumptions.

Findings

Method effectively avoids singularities in practice.

Solutions always satisfy the equality constraints.

Demonstrated effectiveness through two example applications.

Abstract

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.