Abstract
Traditional mathematical programming solvers require long computational times to solve constrained minimization problems of complex and large-scale physical systems. Therefore, these problems are often transformed into unconstrained ones, and solved with computationally efficient optimization approaches based on first-order information, such as the gradient descent method. However, for unconstrained problems, balancing the minimization of the objective function with the reduction of constraint violations is challenging. We consider the class of time-dependent minimization problems with increasing (possibly) nonlinear and non-convex objective function and non-decreasing (possibly) nonlinear and non-convex inequality constraints. To efficiently solve them, we propose a penalty-based guardrail algorithm (PGA). This algorithm adapts a standard penalty-based method by dynamically updating the right-hand side of the constraints with a guardrail variable which adds a margin to prevent violations. We evaluate PGA on two novel application domains: a simplified model of a district heating system and an optimization model derived from learned deep neural networks. Our method significantly outperforms mathematical programming solvers and the standard penalty-based method, and achieves better performance and faster convergence than a state-of-the-art algorithm (IPDD) within a specified time limit.
Abstract (translated)
传统数学编程求解器需要花费长的时间来求解复杂和大规模物理系统的约束最小化问题。因此，通常将这些问题转化为无约束问题，并使用计算效率高的优化方法（如梯度下降法）来求解。然而，对于无约束问题，平衡最小化目标函数与减少约束违反之间的平衡具有挑战性。我们考虑具有增加（可能）非线性和非凸目标函数以及减少（可能）非线性和非凸不等式约束的时间依赖最小化问题。为了有效地解决这些问题，我们提出了一个基于惩罚的守护算法（PGA）。该算法通过动态更新约束的右端随约束惩罚变量的增加，将标准的惩罚方法进行了适应。我们在两个新的应用领域上评估了PGA：一个简化的区域供暖系统的模型和一个学习深度神经网络得到的优化模型。我们的方法显著超过了数学编程求解器和标准惩罚方法，并且在指定的时间限制内实现了更好的性能和更快的收敛速度。
URL
https://arxiv.org/abs/2405.01984
