Discretization-based algorithms for generalized semi-infinite and bilevel programs with coupling equality constraints
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Discretization-based algorithms are proposed for the global solution of mixed-integer nonlinear generalized semi-infinite (GSIP) and bilevel (BLP) programs with lower-level equality constraints coupling the lower and upper level. The algorithms are extensions, respectively, of the algorithm proposed by Mitsos and Tsoukalas (J Glob Optim 61(1):1–17, 2015. https://doi.org/10.1007/s10898-014-0146-6) and by Mitsos (J Glob Optim 47(4):557–582, 2010. https://doi.org/10.1007/s10898-009-9479-y). As their predecessors, the algorithms are based on bounding procedures, which achieve convergence through a successive discretization of the lower-level variable space. In order to cope with convergence issues introduced by coupling equality constraints, a subset of the lower-level variables is treated as dependent variables fixed by the equality constraints while the remaining lower-level variables are discretized. Proofs of finite termination with \(\varepsilon \)-optimality are provided under appropriate assumptions, the preeminent of which are the existence, uniqueness, and continuity of the solution to the equality constraints. The performance of the proposed algorithms is assessed based on numerical experiments.
KeywordsGSIP Bilevel Equality constraints MINLP Nonconvex Global optimization
We gratefully acknowledge the financial support provided by Réseau de transport d’électricité (RTE, France) through the project “Bilevel Optimization for Worst-case Analysis of Power Grids” and by the Excellence Initiative of the German federal state governments through the Cluster of Excellence “Tailor Made Fuels from Biomass” (EXC 236). Moll Glass is grateful for her scholarship from the German Academic Scholarship Foundation (Studienstiftung des deutschen Volkes).
- 22.Kleniati, P.-M., Adjiman, C.S.: Branch-and-sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part II: convergence analysis and numerical results. J. Glob. Optim. 60(3), 459–481 (2014). https://doi.org/10.1007/s10898-013-0120-8 MathSciNetzbMATHGoogle Scholar
- 24.Lemonidis, P.: Global optimization algorithms for semi-infinite and generalized semi-infinite programs. Ph.D. thesis, Massachusetts Institute of Technology, Boston, MA (2008)Google Scholar
- 25.Liu, Z., Gong, Y.-H.: Semi-infinite quadratic optimisation method for the design of robust adaptive array processors. IEE Proc. F 137(3), 177–182 (1990)Google Scholar
- 34.Oluwole, O.O., Barton, P.I., Green, W.H.: Obtaining accurate solutions using reduced chemical kinetic models: a new model reduction method for models rigorously validated over ranges. Combust. Theor. Model. 11(1), 127–146 (2007). https://doi.org/10.1080/13647830600924601 MathSciNetzbMATHGoogle Scholar
- 40.Rosenthal, R.E.: GAMS—a user’s guide. Technical report, GAMS Development Corporation, Washington, DC (2017)Google Scholar
- 57.Weistroffer, V., Mitsos, V.: Relaxation-based bounds for GSIPs. In: Parametric Optimization and Related Topics X (paraoptX), Karlsruhe, Germany (2010)Google Scholar
- 58.Yue, D., Gao, J., Zeng, B., You, F.: A projection-based reformulation and decomposition algorithm for global optimization of mixed integer bilevel linear programs. arXiv:1707.06196v2 (2018)
- 59.Zeng, B., An, Y.: Solving bilevel mixed integer program by reformulations and decomposition. Optim. Online 1–34 (2014)Google Scholar