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.

翻译：传统数学规划求解器在处理复杂大规模物理系统的约束最小化问题时，需要极长的计算时间。因此，这类问题通常被转化为无约束形式，并采用基于一阶信息（如梯度下降法）的计算高效优化方法求解。然而，对于无约束问题，平衡目标函数最小化与约束违反度降低之间的关系具有挑战性。本文考虑一类具有递增（可能为非线性和非凸）目标函数与非递减（可能为非线性和非凸）不等式约束的时间相关最小化问题。为高效求解此类问题，我们提出一种基于罚函数的防护栏算法（PGA）。该算法通过引入动态更新约束右端项的防护栏变量对标准罚函数法进行改进，该变量通过添加裕度来预防约束违反。我们在两个新型应用领域评估了PGA：区域供热系统的简化模型和基于深度神经网络学习的优化模型。实验结果表明，该方法显著优于数学规划求解器和标准罚函数法，并在指定时间限制内比现有最优算法（IPDD）实现了更优性能与更快收敛速度。