将双层参数优化问题几何重构为单层非线性规划问题及其在相平衡中的应用
A geometric reformulation of the bilevel parameter optimization problem to a single level non-linear programming problem with applications to phase equilibria
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中文总结 AI 辅助
针对化学工程中相平衡相关的双层优化问题，本文提出将其几何重构为单层非线性规划的方法，该方法可全局最优求解且适用于各类黑箱热力学模型，为计算热力学难题提供了可靠高效的建模途径。
中文摘要 AI 辅助
相平衡问题是化学工程的核心，支撑着从分离工艺设计到热力学模型开发等各类任务。生成相包络是一项极具挑战性的计算任务：要使热力学模型拟合真实数据并获得可靠预测，需求解计算成本高昂的双层优化问题。本文对该参数估计问题进行几何重构，将双层规划重新表述为更易求解的单层问题。当使用专用全局优化求解器时，该单层问题的解被证明是双层问题的全局最优解。此外，该方法保留了确保拟合模型行为良好的约束，如强制正确的相分裂数量、排除虚假相、在不稳定区域确保稳定性。这使从业者能可靠且高效地将数学复杂的热力学模型拟合到数据，并可能实现对计算热力学中此前难以处理的问题的高精度、严格建模。最后，本文提出一种算法，该算法被证明对任何黑箱热力学模型均收敛，仅需吉布斯自由能表达式，无需导数，且对最广泛的非光滑、非连续模型类保证收敛。
英文摘要
Phase equilibrium problems are central to chemical engineering, underpinning tasks ranging from separation process design to the development of thermodynamic models. A particularly challenging computational task is the generation of phase envelopes: rigorously fitting thermodynamic models to real world data with well behaved predictions requires solving a computationally expensive bilevel optimization problem. We present a geometric reformulation of this parameter estimation problem that restates the bilevel program as a single level problem that is significantly easier to solve. The solution of the single level problem is proven to be the globally optimal solution of the bilevel problem when specialized global optimization solvers are used. In addition, the method retains the constraints that guarantee a well behaved fitted model, such as enforcing the correct number of phase splits, excluding spurious phases, and ensuring stability in regions of instability. This allows the practitioner to reliably and efficiently fit mathematically complex thermodynamic models to data, and potentially enables highly accurate and rigorous modelling of problems in computational thermodynamics that were previously intractable. Finally, an algorithm is presented that is proven to converge for any black box thermodynamic model. Only an expression of the Gibbs free energy is required, no derivatives are needed, and convergence is guaranteed for the broadest class of non-smooth, non-continuous models.
