Abstract
We propose a deterministic global optimization algorithm for mixed-integer nonlinear bilevel problems (MINBP) by generalizing the Branch-and-Sandwich algorithm (Kleniati and Adjiman, 2014a). Advances include the removal of regularity assumptions and the extension of the algorithm to mixed-integer problems. The proposed algorithm can solve very general MINBP problems to global optimality, including problems with inner equality constraints that depend on the inner and outer variables. Inner lower and inner upper bounding problems are constructed to bound the inner optimal value function and provide constant-bound cuts for the outer bounding problems. To remove the need for regularity, we introduce a robust counterpart approach for the inner upper bounding problem. Branching is allowed on all variables without distinction by keeping track of refined partitions of the inner space for every refined subdomain of the outer space. Finite ɛ-convergence to the global solution is proved. The algorithm is applied successfully to 10 mixed-integer literature problems.
Highlights
Many of the optimization problems relevant to chemical engineering are bilevel problems in nature (Grossmann and Biegler, 2004; Gümüsand Floudas, 2005; Floudas and Gounaris, 2009) and can be framed as two-person, hierarchical optimization problems having a second optimization problem as part of the constraints
We present the formulation of the subproblems used in obtaining bounds on the inner and outer problems by considering a node k ∈ Lp in the branch-and-bound tree
The reader is reminded that in the present paper, we propose a different inner upper bounding problem from the one presented in Kleniati and Adjiman (2014a)
Summary
Many of the optimization problems relevant to chemical engineering are bilevel problems in nature (Grossmann and Biegler, 2004; Gümüsand Floudas, 2005; Floudas and Gounaris, 2009) and can be framed as two-person, hierarchical optimization problems having a second optimization problem as part of the constraints. Examples of such formulations include design under uncertainty (Floudas et al, 2001; Ryu et al, 2004), the estimation of parameters in thermodynamic models (Mitsos et al, 2009), the optimization of processes involving phase equilibrium (Kamath et al, 2010) and/or chemical equilibrium (Clark and Westerberg, 1990; Sahin et al, 1998), simultaneous process optimization and heat integration (Kamath et al, 2012), and strategies for product pricing and marketing in competitive markets (Lemonidis, 2008)
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