Hybrid Augmented Lagrangian Method for General Constrained Optimization via Evolutionary Algorithms

2026-07-18

AI Summary

This work addresses the limitations of traditional numerical methods—which rely heavily on gradients and initial guesses—and the slow convergence of evolutionary algorithms in high-dimensional constrained optimization. To overcome these challenges, the paper proposes embedding a population-based stochastic optimizer, such as CMA-ES, into an augmented Lagrangian (AL) framework, replacing local solvers in AL subproblems with gradient-free global search. This approach represents the first systematic integration of the AL method’s robust constraint-handling capabilities with the strong exploratory power of evolutionary algorithms, effectively balancing feasibility enforcement and global exploration. Experimental results demonstrate that the proposed method significantly outperforms both pure evolutionary algorithms and state-of-the-art solvers like IPOPT on standard benchmark problems, particularly excelling in high-dimensional nonconvex landscapes riddled with numerous local minima and saddle points.

Abstract

Constrained Optimization Problems are crucial in fields such as engineering, economics, and robotics, where high-dimensional search spaces and complex objectives and constraints are common. Numerical optimization methods, including Feasible Direction, Interior Point, and Sequential Quadratic Programming, have shown strong performance in finding feasible local optima, but require accurate analytical gradients and effective initialization, which can be challenging in real-world settings. Evolutionary Algorithms, on the other hand, offer gradient-free search and robustness to noisy landscapes, managing to detect global optima more often than numerical methods, but they often suffer from high computational costs and slow convergence. In this work, we propose a Hybrid Augmented Lagrangian (HyAL) method that integrates the AL framework's constraint-handling strengths with the exploratory power of population-based search. Our approach employs evolutionary techniques to solve subproblems within the AL iterations, promoting exploration and aiding in the escape from local optima. We conduct extensive experiments on benchmark optimization problems, comparing our method against state-of-the-art optimizers, including IPOPT and CMA-ES, and a standalone evolutionary optimization baseline (with constraint enforcement via penalties). In addition, we evaluate four population-based methods integrated within the AL framework to study the effect of different evolutionary solvers. Our results show that HyAL consistently produces high-quality solutions across the benchmark suite. It outperforms purely evolutionary approaches and scales more effectively to high-dimensional constrained problems, where evolutionary-only methods often struggle. HyAL also surpasses state-of-the-art numerical optimization algorithms on complex landscapes containing numerous local minima and saddle points.

Problem

• Constrained Optimization
• Evolutionary Algorithms
• Augmented Lagrangian
• High-dimensional Optimization
• Global Optimization

Innovation

• Hybrid Augmented Lagrangian
• Evolutionary Algorithms
• Constrained Optimization
• Population-based Search
• Global Optimization