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A genetic algorithm based augmented Lagrangian method for constrained optimization

机译：基于遗传算法的增强拉格朗日方法约束优化

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摘要

Among the penalty based approaches for constrained optimization, augmented Lagrangian (AL) methods are better in at least three ways: (i) they have theoretical convergence properties, (ii) they distort the original objective function minimally, thereby providing a better function landscape for search, and (iii) they can result in computing optimal Lagrange multiplier for each constraint as a by-product. Instead of keeping a constant penalty parameter throughout the optimization process, these algorithms update the parameters (called multipliers) adaptively so that the corresponding penalized function dynamically changes its optimum from the unconstrained minimum point to the constrained minimum point with iterations. However, the flip side of these algorithms is that the overall algorithm requires a serial application of a number of unconstrained optimization tasks, a process that is usually time-consuming and tend to be computationally expensive. In this paper, we devise a genetic algorithm based parameter update strategy to a particular AL method. The proposed strategy updates critical parameters in an adaptive manner based on population statistics. Occasionally, a classical optimization method is used to improve the GA-obtained solution, thereby providing the resulting hybrid procedure its theoretical convergence property. The GAAL method is applied to a number of constrained test problems taken from the evolutionary algorithms (EAs) literature. The number of function evaluations required by GAAL in most problems is found to be smaller than that needed by a number of existing evolutionary based constraint handling methods. GAAL method is found to be accurate, computationally fast, and reliable over multiple runs. Besides solving the problems, the proposed GAAL method is also able to find the optimal Lagrange multiplier associated with each constraint for the test problems as an added benefit—a matter that is important for a sensitivity analysis of the obtained optimized solution, but has not yet been paid adequate attention in the past evolutionary constrained optimization studies.
机译：在基于惩罚的约束优化方法中，增强拉格朗日（AL）方法至少在以下三种方面更好：（i）具有理论收敛性，（ii）最小地扭曲了原始目标函数，从而为搜索，以及（iii）它们可以导致为每个约束作为副产品计算最佳拉格朗日乘数。这些算法不是在整个优化过程中都保持恒定的惩罚参数，而是自适应地更新参数（称为乘数），以便相应的惩罚函数通过迭代将其最优值从不受约束的最小点动态更改为受约束的最小点。但是，这些算法的另一面是，整个算法需要一系列无限制的优化任务的串行应用，该过程通常很耗时，并且在计算上往往很昂贵。在本文中，我们针对特定的AL方法设计了一种基于遗传算法的参数更新策略。所提出的策略基于人口统计以自适应方式更新关键参数。有时，使用经典的优化方法来改进遗传算法获得的解，从而为所得的混合过程提供理论上的收敛性。 GAAL方法应用于从进化算法（EA）文献中得出的许多约束测试问题。发现在大多数问题中，GAAL所需的功能评估数量少于许多现有的基于演化的约束处理方法所需的功能评估数量。发现GAAL方法在多次运行中都是准确，计算快速且可靠的。除了解决问题外，拟议的GAAL方法还能够为测试问题找到与每个约束相关的最佳Lagrange乘数，以作为一项额外的好处-对于获得的优化解决方案的灵敏度分析而言，这一点很重要，但尚未解决在过去的演化约束优化研究中已得到足够的重视。

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