A Second-Order Descent Algorithm for Nonlinear Equality Constrained Optimization

Abstract

In this paper, an adaptive second-order descent algorithm (ASODA) is proposed for solving nonlinear equality constrained optimization problems. By integrating a sequential quadratic programming approach with the homogeneous second-order descent framework (HSODF: He et al., Math. Program. 215, 575--636, 2026), each iteration of ASODA decomposes the trial step into a normal step and a tangential step. The latter is computed by converting the linearly constrained subproblem into an unconstrained one via the reduced-Hessian technique and then solving it using a generalized homogeneous model. The algorithm employs a merit function to evaluate step acceptability. Under standard smoothness assumptions, we establish a global iteration complexity analysis for ASODA toward an $ε$-approximate second-order stationary point; the iteration count scales as $\mathcal{O}(ε^{-3/2})$ up to constants whose dependence on $ε$ is made explicit in Section 3; the bound is stated there under an explicit structural hypothesis on the regularization parameter ($κ_{\rm h}=O(1)$), which we keep visible rather than treating any constant as silently $ε$-independent. This matches the optimal complexity bound achieved by HSODF for unconstrained problems. Numerical experiments on CUTEst problems demonstrate that ASODA often requires significantly fewer function and gradient evaluations than existing second-order methods, confirming its effectiveness and competitiveness.