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This paper considers the regularization continuation method and thetrust-region updating strategy for the optimization problem with linearequality constraints.The proposed method utilizes the linear conservation lawof the regularization continuation method such that it does not need to computethe correction step for preserving the feasibility other than the previouscontinuation methods and the quasi-Newton updating formulas for the linearlyconstrained optimization problem. Moreover, the new method uses the speciallimited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) formula as thepreconditioning technique to improve its computational efficiency in thewell-posed phase, and it uses the inverse of the regularized two-sidedprojection of the Lagrangian Hessian as the pre-conditioner to improve itsrobustness. Numerical results also show that the new method is more robust andfaster than the traditional optimization method such as the alternatingdirection method of multipliers (ADMM),the sequential quadratic programming(SQP) method (the built-in subroutine fmincon.m of the MATLAB2020aenvironment), and the recent continuation method (Ptctr). The computationaltime of the new method is about 1/3 of that of SQP (fmincon.m). Finally, theglobal convergence analysis of the new method is also given.

The regularization continuation method with an adaptive time step control for linearly constrained optimization problems