In this work we present deep learning implementations of two popular theoretical constrained optimization algorithms in infinite dimensional Hilbert spaces, namely, the penalty and the augmented Lagrangian methods. We test these algorithms on some toy problems originating in either calculus of variations or physics. We demonstrate that both methods are able to produce decent approximations for the test problems and are comparable in terms of different errors produced. Leveraging the common occurrence of the Lagrange multiplier update rule being computationally less expensive than solving subproblems in the penalty method, we achieve significant speedups in cases when the output of the constraint function is itself a function.

翻译：本文提出了两种流行理论约束优化算法——惩罚方法和增广拉格朗日方法——在无限维希尔伯特空间中的深度学习实现。我们在源于变分法或物理学的若干玩具问题上测试了这些算法。实验表明，两种方法均能为测试问题生成合理的近似解，且在不同误差指标上表现相当。利用拉格朗日乘子更新规则在计算成本上通常低于惩罚方法中子问题求解这一常见特性，我们在约束函数输出本身为函数的情形下实现了显著的加速效果。