Scalable First-Order Interior Point Trust Region Algorithms for Linearly Constrained Optimization

2026-04-27

AI Summary

This work addresses the high computational cost of traditional interior-point trust-region methods for constrained nonconvex optimization, which stems from repeatedly solving trust-region subproblems. To overcome this limitation, we propose an approximate first-order interior-point trust-region framework that maintains an approximate projection operator via low-rank updates, thereby avoiding explicit subproblem solves and Hessian evaluations. By integrating a gradient-based negative curvature routine, the method efficiently approximates both first- and second-order Karush–Kuhn–Tucker (KKT) points using only first-order information. Theoretical analysis establishes feasibility and convergence guarantees for the iterates, while numerical experiments demonstrate up to a 2.48× speedup over state-of-the-art methods on large-scale problems, significantly enhancing scalability.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

Computing approximate Karush--Kuhn--Tucker (KKT) points for constrained nonconvex programs is a fundamental problem in mathematical programming. Interior-point trust-region (IPTR) methods are particularly attractive for such problems because they maintain strictly feasible iterates throughout the iterative process and converge to a first-order and second-order KKT solution. Their scalability, however, is limited by the repeated computation of trust-region search directions. In this paper, we propose an approximate first-order IPTR framework that addresses this bottleneck by replacing exact trust-region subproblem solves with an approximate projector maintained through low-rank updates. The resulting method preserves feasibility and the global convergence guarantees of standard IPTR schemes while substantially reducing the per-iteration cost. We further extend the framework to obtain approximate second-order KKT points using only first-order information by integrating a gradient-based negative-curvature routine, thus avoiding explicit Hessian computations. We conduct numerical experiments to demonstrate the scalability of our approximate first-order IPTR framework in large-scale settings, where it achieves up to a $2.48\times$ speedup over the existing first-order IPTR algorithm.

Problem

• constrained nonconvex optimization
• Karush–Kuhn–Tucker points
• interior-point trust-region
• scalability
• linearly constrained optimization

Innovation

• interior-point trust-region
• first-order method
• low-rank approximation
• KKT points
• scalable optimization