A scalable sequential adaptive cubic regularization algorithm for optimization with general equality constraints
Yonggang Pei, Shuai Shao, Mauricio Silva Louzeiro, Detong Zhu
TL;DR
This paper addresses large-scale nonlinear optimization with general equality constraints by extending the ARC_qK framework to a scalable SSARC_qK method. The approach splits each step into a vertical feasibility move and a horizontal objective-reduction move, solving the horizontal ARC subproblem in the null space of the linearized constraints with CG-Lanczos using shifts, and guiding acceptance with an exact-penalty merit. The authors establish a new global convergence analysis under mild assumptions and report preliminary numerical results on CUTEst problems, demonstrating the method's scalability and robustness. The work provides a practical, convergent framework for constrained optimization that leverages reduced-Hessian techniques and iterative Krylov solvers, with potential for further analysis of complexity and local convergence.
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, complexity, and promising numerical performance. In this paper, we extend $\mathrm{ARC_{q}K}$ to large scale 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. Then composite-step approach is used to decompose the trial step into the sum of the vertical step and the horizontal step. By means of reduced-Hessian approach, we rewrite the linearity constrained ARC subproblem as a standard unconstrained ARC subproblem to compute the horizontal step. A CG-Lanczos procedure with shifts is employed to solve this subproblem approximately. We provide a new global convergence analysis of the inexact ARC method. Preliminary numerical results are reported to show the performance of $\mathrm{SSARC_{q}K}$.