A Second-Order Algorithm Based on Affine Scaling Interior-Point Methods for nonlinear Optimisation with bound constraints

TL;DR

This paper introduces SOBASIP, a second-order interior-point method for bound-constrained nonlinear optimization, extending previous unconstrained approaches with theoretical guarantees and efficient eigenvalue-based subproblems.

Contribution

It extends the homogeneous second-order descent method to bound-constrained problems using affine scaling and homogenisation techniques, providing global complexity and local superlinear convergence.

Findings

Achieves a global iteration complexity of O(ε^{-3/2})

Uses eigenvalue problems for efficient subproblem solving

Demonstrates satisfactory numerical performance

Abstract

The homogeneous second-order descent method (Zhang et al. 2025, Mathematics of Operations Research) was initially proposed for unconstrained optimisation problems. HSODM shows excellent performance with respect to the global complexity rate among a certain broad class of second-order methods. In this paper, we extend HSODM to solve nonlinear optimisation problems with bound constraints and propose a second-order algorithm based on affine scaling interior-point methods (SOBASIP). In each iteration, an appropriate affine matrix is introduced to construct an affine scaling subproblem based on the optimality conditions of the problem. To obtain a valid descent direction similar to HSODM, we utilise the homogenisation technique to transform the scaling subproblem into an Ordinary Homogeneous Model (OHM), which is essentially an eigenvalue problem that can be solved efficiently. The descent direction is constructed from the optimal solution to the OHM, and then backtracking line search is used to determine the new iteration point. Theoretical analysis establishes that SOBASIP achieves a global iteration complexity of $O(\epsilon^{-3/2})$ for finding an $\epsilon$-approximate second-order stationary point and converges locally at a superlinear rate under certain conditions. Numerical results demonstrate that the proposed method exhibits satisfactory performance.