We introduce a generalized value function of a mixed-integer program, which is simultaneously parameterized by its objective and right-hand side. We describe its fundamental properties, which we exploit through three algorithms to calculate it. We then show how this generalized value function can be used to reformulate two classes of mixed-integer optimization problems: two-stage stochastic mixed-integer programming and multi-follower bilevel mixed-integer programming. For both of these problem classes, the generalized value function approach allows the solution of instances that are significantly larger than those solved in the literature in terms of the total number of variables and number of scenarios.
Citation
@article{Tavaslioglu2019, author = {Tavasl{\i}o\u{g}lu, O. and Prokopyev, Oleg A. and Schaefer, Andrew J.}, title = {Solving Stochastic and Bilevel Mixed-Integer Programs via a Generalized Value Function}, journal = {Operations Research}, volume = {67}, number = {6}, pages = {1659-1677}, year = {2019}, doi = {10.1287/opre.2019.1842}, URL = { https://doi.org/10.1287/opre.2019.1842 }, eprint = { https://doi.org/10.1287/opre.2019.1842 } , abstract = { We introduce a generalized value function of a mixed-integer program, which is simultaneously parameterized by its objective and right-hand side. We describe its fundamental properties, which we exploit through three algorithms to calculate it. We then show how this generalized value function can be used to reformulate two classes of mixed-integer optimization problems: two-stage stochastic mixed-integer programming and multifollower bilevel mixed-integer programming. For both of these problem classes, the generalized value function approach allows the solution of instances that are significantly larger than those solved in the literature in terms of the total number of variables and number of scenarios. } }