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This paper considers the regularization continuation method and thetrust-region updating strategy for the nonlinearly equality-constrainedoptimization problem. Namely, it uses the inverse of the regularizationquasi-Newton matrix as the pre-conditioner to improve its computationalefficiency in the well-posed phase, and it adopts the inverse of theregularization two-sided projection of the Hessian as the pre-conditioner toimprove its robustness in the ill-conditioned phase. Since it only solves alinear system of equations at every iteration and the sequential quadraticprogramming (SQP) needs to solve a quadratic programming subproblem at everyiteration, it is faster than SQP. Numerical results also show that it is morerobust and faster than SQP (the built-in subroutine fmincon.m of theMATLAB2020a environment and the subroutine SNOPT executed in GAMS v28.2 (2019)environment). The computational time of the new method is about one third ofthat of fmincon.m for the large-scale problem. Finally, the global convergenceanalysis of the new method is also given.

The regularization continuation method for optimization problems with nonlinear equality constraints