On the feasibility of generalized inverse linear programs
Christoph Buchheim, Lowig T. Duer
TL;DR
The paper investigates the generalized inverse linear feasibility problem (InvLFP), where both the objective and the right-hand side of a linear program are affinely parameterized and a target set $Y$ constrains the optimal solution. It provides a comprehensive complexity taxonomy across target types (singleton, basis, polyhedral, and oracle-defined) and LP forms (natural vs. standard), under optimistic and pessimistic scenarios, establishing NP-hardness results, NP-membership, and fixed-parameter tractability for partial targets. Key findings show tractability in standard-form LPs for singleton targets, NP-hardness in several natural-form variants, and differing complexities between optimistic and pessimistic interpretations largely due to degeneracy considerations; polyhedral and oracle-defined targets introduce additional nuanced behavior. The results inform both theoretical understanding and practical design of inverse optimization models, with implications for bilevel reformulations and feasibility certificates in inverse problems.
Abstract
We investigate the feasibility problem for generalized inverse linear programs. Given an LP with affinely parametrized objective function and right-hand side as well as a target set Y, the goal is to decide whether the parameters can be chosen such that there exists an optimal solution that belongs to Y (optimistic scenario) or such that all optimal solutions belong to Y (pessimistic scenario). We study the complexity of this decision problem and show how it depends on the structure of the set Y, the form of the LP, the adjustable parameters, and the underlying scenario. For a target singleton Y = {y}, we show that the problem is tractable if the given LP is in standard form, but NP-hard if the LP is given in natural form. If instead we are given a target basis B, the problem in standard form becomes NP-complete in the optimistic case, while remaining tractable in the pessimistic case. For partially fixed target solutions, the problem gets almost immediately NP-hard, but we prove fixed-parameter tractability in the number of non-fixed variables. Moreover, we give a rigorous proof of membership in NP for any polyhedral target set, and discuss how this property can be extended to more general target sets using an oracle-based approach.