(120a) Bilevel Programs with Coupling Equality Constraints for Parameter Estimation of Thermodynamic Property Models

As early as 1982, bilevel programs and related semi-infinite programs have been considered in chemical engineering, e.g., in the context of process flexibility [1]. However, classical bilevel theory is limited to the solution of linear bilevel programs or at least programs with convex lower-level program.

In recent years, more general bilevel programs with nonconvex lower-level programs have seen considerable activity, both in terms of algorithms [2, 3, 4, 5, 6, 7] and engineering applications. Examples are parameter estimation problems for thermodynamic property models with sufficient thermodynamic stability criteria [8] and the worst case stability analysis of power grids [9]. While real-world problems have been solved, current algorithms still pose restrictions on the problems that can be solved. As such, bilevel programs with equality constraints coupling the lower and upper level pose a significant challenge, either due to presenting an adverse case in terms of computational effort spent by the algorithms, or due to violating assumptions that are required for convergence.

Indeed, discretization methods for bilevel programming derive single-level subproblems by discretizing the lower-level variable space of the bilevel program. Iterative population strategies for those discretizations have been shown to yield convergent bounding algorithms under appropriate assumptions. These usually include an assumption of a Slater-like condition for the lower-level program, which fails to be satisfied in the presence of lower-level equality constraints coupling the lower and upper level.

Based on similar assumptions as related work for semi-infinite programs [10, 11], we propose an extension of an existing discretization method for nonlinear mixed-integer bilevel programs [2, 3] to alleviate that problem and allow lower-level equality constraints coupling the lower and upper level. The method is guaranteed to converge to an ε-optimal solution in finite time under appropriate assumptions. The central assumption is concerned with a uniqueness property of the solution to the equality constraints. Beyond an incrementally increasing number of constraints typically associated with discretization methods, the subproblems of the proposed method exhibit an incremental increase in the number of variables.

The viability of the extended method is assessed based on the solution of parameter estimation problems for thermodynamic property models with the equality constraints stemming from cubic equations of state.

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