A Unified Framework for Multistage Mixed Integer Linear Optimization

Abstract

We introduce a unified framework for the study of multilevel mixed integer linear optimization problems and multistage stochastic mixed integer linear optimization problems with recourse. The framework highlights the common mathematical structure of the two problems and allows for the development of a common algorithmic framework. Focusing on the two-stage case, we investigate, in particular, the nature of the value function of the second-stage problem, highlighting its connection to dual functions and the theory of duality for mixed integer linear optimization problems, and summarize different reformulations. We then present two main solution techniques, one based on a Benders-like decomposition to approximate either the risk function or the value function, and the other one based on cutting plane generation.

Airport capacity extension, fleet investment, and optimal aircraft scheduling in a multilevel market model: quantifying the costs of imperfect markets

“…Assuming rational players, the optimal decisions made by a player should therefore take into account how the other players would react to it, factoring their reaction into the player's own decisionmaking process so to choose a strategy which is best possible for her/him/it. Similar assumptions are made in the operations research and mathematical programming literature on bilevel (and multilevel) optimization, see Colson et al (2007), Bolusani et al (2020) for a survey, and are rooted in the game-theoretical literature on Stackelberg and hierarchical games, whose origin is in Von Stackelberg (1934).…”

“…See, for instance, (Dempe 2002;Colson et al 2007;Dempe et al 2014;Coniglio et al 2017;Basilico et al 2017Basilico et al , 2020Coniglio et al 2020) in which hardness and inapproximability results are shown for simpler problems featuring only two levels and a single player per level. For a recent survey, we refer the reader to Bolusani et al (2020). To find a solution to the model we proposed, in this paper we develop a reformulation strategy which builds on the steps depicted in Fig.…”

Optimizing over the Closure of Rank Inequalities with a Small Right-Hand Side for the Maximum Stable Set Problem via Bilevel Programming

“…Fore more details on the relationship between cutting plane generation, bilevel programming, and the polynomial hierarchy, we refer the reader to (Lodi et al 2014). For a recent survey on mixed integer multilevel programming (and its relationship with mixed integer multistage programming), we refer the reader to (Bolusani et al 2020).…”

Complexity of near-optimal robust versions of multilevel optimization problems

“…The linear bilevel problem corresponds to s = 1 and is in Σ P 1 ≡ N P. If, on the contrary, at least the bottom-level problem involves integrality constraints (or more generally belongs to N P but not P), the multilevel problem with s levels belongs to Σ P s . A model unifying multistage stochastic and multilevel problems is defined in [18], based on a risk function capturing the component of the objective function which is unknown to a decision-maker at their stage. Complexity and completeness results in the polynomial hierarchy above the first level are compiled in [19].…”

“…As highlighted in [18], most results in the literature on complexity of multilevel optimization use N P-hardness as the sole characterization. This only indicates that a given problem is at least as hard as all problems in N P and that no polynomial-time solution method should be expected unless N P = P.…”