Jul 2026· Proceedings of the Genetic and Evolutionary Computation Conference Companion· 0 citations· 21 references
TL;DR
This work proposes a Hybrid Augmented Lagrangian (HyAL) method that combines the constraint-handling capabilities of the AL framework with the exploratory power of population-based search, and demonstrates that HyAL effectively solves constrained optimization problems, including high-dimensional cases where population-based methods struggle.
Abstract
Constraint Optimization is crucial to fields such as engineering, economics, and robotics, where high-dimensional search spaces and complex constraints are common. Numerical optimization methods, like Interior Point and Sequential Quadratic Programming, achieve strong performance but rely on accurate gradients and good initialization, which are difficult to obtain. Evolutionary Algorithms (EAs) provide gradient-free search and robustness to complex landscapes, but suffer from high computational cost and slow convergence. In this work, we propose a Hybrid Augmented Lagrangian (HyAL) method that combines the constraint-handling capabilities of the AL framework with the exploratory power of population-based search. Evolutionary techniques are used to solve subproblems within the AL iterations, enhancing exploration and enabling escape from local optima. We evaluate four population-based methods within this framework and compare them against a state-of-the-art solver. Results demonstrate that HyAL effectively solves constrained optimization problems, including high-dimensional cases where population-based methods struggle.
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.
Lampros Printzios, Konstantinos I. Chatzilygeroudis· 0 citations
This study introduces a novel constrained multi-objective evolutionary algorithm, termed DPCME, which employs two interacting populations that exchange information, enabling effective global exploration and reducing the risk of convergence to local optima.
A unified and comprehensive perspective on constructing augmented Lagrangian functions (based on the Hestenes–Powell–Rockafellar augmented Lagrangian) for various optimization problems, including nonlinear programming and convex and nonconvex composite programming.
Nonlinear Continuous global Optimization Problems (NCOPs) are well-known problems that arise in many applications, from engineering to robotics. The Branch & Bound method is a widely used approach for solving NCOPs to global optimality, often interleaving techniques like bisection and filtering. A key aspect of this approach is identifying feasible solutions early in the search process, which enables effective pruning of the search tree and avoids unnecessary computational effort. Inner linear relaxation techniques, such as the AbsTaylor strategy, have proven effective for identifying feasible regions; however, they heavily rely on a heuristically chosen expansion point (often the box midpoint), which directly impacts solution quality and relaxation success. In this work, we propose a novel gradient-based strategy to dynamically optimize the selection of this expansion point. By employing Gradient Descent to minimize a Mean Squared Error (MSE) objective formulated exclusively over the active constraints, we systematically guide the expansion point safely away from boundaries and into a strictly feasible interior region. To manage computational overhead, we evaluate restricted iteration budgets alongside algorithmic variants, specifically introducing a point inheritance strategy for warm-starting and comparing Batch versus Incremental gradient updates. Experimental results on a well-known benchmark set demonstrate that this gradient-based approach minimizes the probability of relaxation failure, significantly enhancing pruning effectiveness and overall solver efficiency compared to the original strategy.
For solving nonconvex equality-constrained optimization problems, a recent Gradient-Eigenstep Algorithm by Goyens et al.~is an iteration-efficient approach, based on minimizing Fletcher's augmented Lagrangian function, for finding an approximate second-order stationary point from an arbitrary starting point. In this paper, the analysis of this algorithm is extended, offering a two-fold contribution. First, it is shown that a local-linear rate of convergence can be obtained by this method if it is initiated sufficiently close to a strong second-order stationary point and employs a sufficiently small step-size parameter and sufficiently large penalty parameter. In this case, the algorithm reduces to a gradient descent algorithm applied to minimize Fletcher's augmented Lagrangian. Second, as a particularly useful application of the first result, it is shown that the Gradient-Eigenstep algorithm can be used as an iteration-efficient subproblem solver in the context of a progressive sampling strategy for solving equality-constrained optimization problems when the objective and constraint functions are defined by large sample averages, ultimately offering an algorithm with an improved worst-case sample complexity when compared to an approach that solves a full-sample problem directly.
Frank E. Curtis, Ling-Jun Guo, Daniel P. Robinson· 0 citations
Results show that integrating local search significantly enhances performance, while a principled method for setting hybrid parameters ensures robustness and reproducibility, highlighting the potential of combining mathematical programming techniques with evolutionary algorithms for high-dimensional many-objective optimization problems.
Regina C. L. C. de Sousa, Dênis E. C. Vargas, Elizabeth F. Wanner et al.· Journal of Heuristics· 0 citations