The scalable adaptive cubic regularization method ( \(\varvec{\textrm{ARC}}_{\varvec{\textrm{q}}}\varvec{\textrm{K}}\) : [1]) has been recently proposed for unconstrained optimization. It has excellent convergence properties, well-defined complexity bounds, and promising numerical performance. In this paper, we extend \(\varvec{\textrm{ARC}}_{\varvec{\textrm{q}}}\varvec{\textrm{K}}\) to nonlinear optimization with general equality constraints and propose a scalable sequential adaptive cubic regularization algorithm named \(\varvec{\textrm{SSARC}}_{\varvec{\textrm{q}}}\varvec{\textrm{K}}\) . In each iteration, we construct an ARC subproblem with linearized constraints inspired by sequential quadratic optimization methods. Next, a composite-step approach is used to decompose the trial step into the sum of a vertical step and a horizontal step. By means of the reduced-Hessian approach, we rewrite the linearly constrained ARC subproblem as a standard unconstrained ARC subproblem to compute the horizontal step. Analogous to \(\varvec{\textrm{ARC}}_{\varvec{\textrm{q}}}\varvec{\textrm{K}}\) , we employ a CG-Lanczos procedure with shifts to solve ARC subproblems inexactly, thus bypassing any hard case consideration. This also avoids solving the subproblem multiple times for obtaining a new iterative point. We establish the global convergence of the inexact ARC method \(\varvec{\textrm{SSARC}}_{\varvec{\textrm{q}}}\varvec{\textrm{K}}\) to first-order critical points. Preliminary numerical tests and some comparison results are presented to illustrate the performance of \(\varvec{\textrm{SSARC}}_{\varvec{\textrm{q}}}\varvec{\textrm{K}}\) .
