The Regularization Continuation Method for Optimization Problems with Nonlinear Equality Constraints

This paper considers the regularization continuation method and the trust-region updating strategy for the nonlinearly equality-constrained optimization problem. Namely, it uses the inverse of the regularization quasi-Newton matrix as the pre-conditioner and the Sherman-Morrison-Woodbury formula to improve its computational efficiency in the well-posed phase, and it adopts the inverse of the regularization two-sided projection of the Hessian as the pre-conditioner to improve its robustness in the ill-conditioned phase. Since it only solves a linear system of equations and the Sherman-Morrison-Woodbury formula significantly saves the computational time at every iteration, and the sequential quadratic programming (SQP) method needs to solve a quadratic programming subproblem at every iteration, it is faster than SQP. Numerical results also show that it is more robust and faster than SQP (the built-in subroutine fmincon.m of the MATLAB2020a environment and the subroutine SNOPT executed in GAMS v28.2 (GAMS Corporation, 2019) environment). The computational time of the new method is about one third of that of fmincon.m for the large-scale problem. Finally, the global convergence analysis of the new method is also given.