Optimality conditions for mixed discrete bilevel optimization problems

Abstract

Usually, bilevel optimization problems need to be transformed into singlelevel ones in order to derive optimality conditions and solution algorithms. Among the available approaches, the replacement of the lower-level problem by means of duality relations became popular quite recently. We revisit three realizations of this idea which are based on the lower-level Lagrange, Wolfe, and Mond-Weir dual problem. The resulting single-level surrogate problems are equivalent to the original bilevel optimization problem from the viewpoint of global minimizers under mild assumptions. However, all these reformulations suffer from the appearance of so-called implicit variables, i.e., surrogate variables which do not enter the objective function but appear in the feasible set for modeling purposes. Treating implicit variables as explicit ones has been shown to be problematic when locally optimal solutions, stationary points, and applicable constraint qualifications are compared to the original problem. Indeed, we illustrate that the same difficulties have to be faced when using these duality-based reformulations. Furthermore, we show that the Mangasarian-Fromovitz constraint qualification is likely to be violated at each feasible point of these reformulations, contrasting assertions in some recently published papers.

A Review on Bilevel Optimization: From Classical to Evolutionary Approaches and Applications

“…In this approach, the authors place a cut, similar to the one used by Bialas and Karwan [26] (for the continuous linear bilevel program), seeking incremental improvements in the upper level objective function. A cutting plane method utlizing the Chvátal-Gomory cut for the continuous-discrete bilevel program was proposed in [60]. Benders-decomposition based techniques have also been employed to solve bilevel problems with mixed integers at the upper level and continuous linear programs at the lower level.…”

A projection-based reformulation and decomposition algorithm for global optimization of a class of mixed integer bilevel linear programs

“…Moore and Bard [8,26] proposed the first branch-and-bound algorithms for MIBLPs. Dempe [19] and Hemmati and Smith [16] proposed a cutting plane approach. Saharidis and Ierapetritou [27] proposed an algorithm based on Benders decomposition.…”

“…However, the detailed model formulation and the resulting problem size of this MIBLP are not reported. Most existing MIBLP algorithms are proposed to handle special classes of (P0), such as integer bilevel linear programs [15], MIBLPs with special constraint structures [16], MIBLPs without continuous upper-level variables [7,17,18], and/or MIBLPs without continuous lowerlevel variables [19,20]. Fischetti et al [21,22] introduced a new general-purpose algorithm for MIBLPs based on a branch-and-cut framework, where new classes of valid inequalities and effective preprocessing procedures are introduced.…”

Model-based multi-parametric programming strategies towards the integration of design, control and operational optimization

“…Simple Tabu search [302] Simulated Annealing [86] Polynomial Approximation -Cutting plane [87] Penalty Function [252] Genetic Algorithm [88] Branch and Bound -Cutting plane [301] Benders decomposition [185] Parametric integer programming [363] Branch and Bound -Alg BM ILP…”