A scalable sequential adaptive cubic regularization algorithm for optimization with general equality constraints

Abstract

The scalable adaptive cubic regularization method ($\mathrm{ARC_{q}K}$: Dussault et al. in Math. Program. Ser. A 207(1-2): 191-225, 2024) 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 $\mathrm{ARC_{q}K}$ to nonlinear optimization with general equality constraints and propose a scalable sequential adaptive cubic regularization algorithm named $\mathrm{SSARC_{q}K}$. 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 $\mathrm{ARC_{q}K}$, 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 $\mathrm{SSARC_{q}K}$ to first-order critical points. Preliminary numerical tests and some comparison results are presented to illustrate the performance of $\mathrm{SSARC_{q}K}$.