Solving Stochastic and Bilevel Mixed-Integer Programs via a Generalized Value Function

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.

Managing Product Transitions: A Bilevel Programming Approach

“…[11] formulate a rank pricing problem as a nonlinear bilevel program with multiple independent followers and develop a tailored branch-and-cut algorithm. [44] apply a generalized value function, whose arguments are the objective function gradient and right-hand-side of constraints, to a multiple-follower MIBLP whose followers share a common constraint matrix. [8] study a BLP whose multiple followers play a Nash game, establishing several structural properties and proposing a branch-and-bound method.…”

A Gilmore-Gomory-Type Construction of Integer Programming Value Functions

“…Various value function approaches have also been applied to the solution of mixed-integer bilevel programs. [13] use a generalized MIP value function, which takes both the objective function coefficients and the constraint right-hand side as arguments, to solve both stochastic and multifollower bilevel MIPs. [3] use Gomory, Chvátal, and Jeroslow functions, as well as value functions, to analyze the representability of mixed-integer bilevel programs.…”

Mixed-integer bilevel representability

“…Indeed, it is the non-closed nature of generalized polyhedra that allows us to study the non-closed feasible regions of MIBL-programs. Specifically, these tools arise when we take the value function approach to bilevel programming, as previously studied in [16,25,34,37]. Here, we leverage the characterization of Blair [8] of the value function of the mixed-integer program in the lower level problem (2).…”

“…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.…”