Adaptive discretization-based algorithms for semi-infinite programs with unbounded variables

Abstract

The proof of convergence of adaptive discretization-based algorithms for semi-infinite programs (SIPs) usually relies on compact host sets for the upper- and lower-level variables. This assumption is violated in some applications, and we show that indeed convergence problems can arise when discretization-based algorithms are applied to SIPs with unbounded variables. To mitigate these convergence problems, we first examine the underlying assumptions of adaptive discretization-based algorithms. We do this paradigmatically using the lower-bounding procedure of Mitsos [Optimization 60(10–11):1291–1308, 2011], which uses the algorithm proposed by Blankenship and Falk [J Optim Theory Appl 19(2):261–281, 1976]. It is noteworthy that the considered procedure and assumptions are essentially the same in the broad class of adaptive discretization-based algorithms. We give sharper, slightly relaxed, assumptions with which we achieve the same convergence guarantees. We show that the convergence guarantees also hold for certain SIPs with unbounded variables based on these sharpened assumptions. However, these sharpened assumptions may be difficult to prove a priori. For these cases, we propose additional, stricter, assumptions which might be easier to prove and which imply the sharpened assumptions. Using these additional assumptions, we present numerical case studies with unbounded variables. Finally, we review which applications are tractable with the proposed additional assumptions.

Itaconic acid production by co‐feeding of Ustilago maydis: A combined approach of experimental data, design of experiments, and metabolic modeling

“…The upper level of the optimization program describes the bioengineering objective to maximize the flux of the target product, whereas the lower level describes the microorganism that aims to maximize the biomass flux. The reformulated form of OptKnock as a single‐level program was implemented in the optimization language libALE (Djelassi & Mitsos, 2019) and solved using libDIPS (Jungen et al, 2023) with gurobi 9.5.2 (Gurobi Optimization, LLC, 2022) as solver. The code for the OptKnock computations is openly available (https://github.com/iAMB-RWTH-Aachen/Ustilago_maydis-GEM/tree/master/data/AcetateCofeed).…”

“…The reformulated form of OptKnock as a single-level program was implemented in the optimization language libALE (Djelassi & Mitsos, 2019) and solved using libDIPS (Jungen et al, 2023) with gurobi 9.5.2 (Gurobi Optimization, LLC, 2022) as solver. The code for the OptKnock computations is openly available (https://github.com/iAMB-RWTH-Aachen/Ustilago_maydis-GEM/tree/master/data/AcetateCofeed).…”

Unified Flash Calculations with Isenthalpic and Isochoric Constraints

“…The initialization procedure can also be replaced by arguments based on the duality principle [31], where the solution of the dual problem provides all the information necessary to reconstruct all stable phases in a given thermodynamic state. It is possible to solve such problems numerically using the procedures given by Jungen et al [32]. In spite of this, such approaches are outside the scope of the present study and are left for future investigation.…”

Itaconic acid production by co-feeding ofUstilago maydis: a combined approach of experimental data, design of experiments and metabolic modeling

“…The upper level of the optimization program describes the bioengineering objective to maximize the flux of the target product, whereas the lower level describes the microorganism that aims to maximize the biomass flux. The reformulated form of OptKnock as a single-level program was implemented in the optimization language libALE [37] and solved using libDIPS [38] with gurobi 9.5.2 [39] as solver. The code for the OptKnock computations is openly available (https://github.com/iAMB-RWTH-Aachen/ Ustilago_maydis-GEM/tree/master/data/AcetateCofeed).…”