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

Abstract

We propose an extended variant of the reformulation and decomposition algorithm for solving a special class of mixed-integer bilevel linear programs (MIBLPs) where continuous and integer variables are involved in both upper-and lower-level problems. In particular, we consider MIBLPs with upper-level constraints that involve lower-level variables. We assume that the inducible region is nonempty and all variables are bounded. By using the reformulation and decomposition scheme, an MIBLP is first converted into its equivalent single-level formulation, then computed by a column-and-constraint generation based decomposition algorithm. The solution procedure is enhanced by a projection strategy that does not require the relatively complete response property. To ensure its performance, we prove that our new method converges to the global optimal solution in a finite number of iterations. A large-scale computational study on random instances and instances of hierarchical supply chain planning are presented to demonstrate the effectiveness of the algorithm.

Discretization-based algorithms for generalized semi-infinite and bilevel programs with coupling equality constraints

“…Zeng and An [59] consider mixed-integer linear BLPs with upper-level constraints only depending on upper-level variables and propose a decomposition algorithms for their solution. The approach is extended to include upperlevel constraints depending on lower-level variables by Yue et al [58]. Hemmati and Smith [17] propose a cutting-plane algorithm for a mixed-integer linear BLP of particular structure.…”

Mixed-integer bilevel representability

“…Bilevel programming has a long history, with traditions in theoretical economics (see, for instance, [29], which originally appeared in 1975) and operations research (see, for instance, [10,22]). While much of the research community's attention has focused on the continuous case, there is a growing literature on bilevel programs with integer variables, starting with early work in the 1990s by Bard and Moore [3,30] through a more recent surge of interest [16,18,19,24,25,33,34,36,38]. Research has largely focused on algorithmic concerns, with a recent emphasis on leveraging advancements in cutting plane techniques.…”

Αλγόριθμοι Διεπίπεδου Προγραμματισμού Για Βέλτιστη Υποβολή Προσφορών Παραγωγών Ενέργειας Σε Αγορές Ημερήσιου Προγραμματισμού Ηλεκτρικής Ενέργειας

“…Yue et al (2019) [63] present a reformulation and decomposition algorithm for mixed integer bilevel linear programming. The proposed algorithm implements a column and constraint generation methodology utilizing a master problem and suitable subproblems on a projection-based single-level problem reformulation.…”

“…Considerable computational difficulties are also inherent in methodologies employing KKT techniques such as the ones by Gümüs and Floudas (2005) [24], Mitsos (2010) [46], and Yue et al (2019) [63], since they necessitate the introduction of dual variables as well as big-M formulations for the treatment of the associated complementary slackness constraints. This is yet another factor that may introduce intolerable obstacles in large realistic problems.…”

“…The reformulation algorithm of Yue et al ( 2019) [63] necessitates the introduction of a number of constraints which grows exponentially with the number of lower-level integer variables. This may render its application on large scale realistic problems intractable.…”