arXiv subjects

Diego Moran Ramirez

Publications and source records attributed to Diego Moran Ramirez.

3 recordsLinked to original sources

arXiv2026

On the integrality gap of convex mixed-integer programs

We study the integrality gap of convex mixed-integer programs, that is, the difference between the optimal value of such a problem and the optimal value of its continuous relaxation. We study classes of convex sets whose associated optimization problem have finite integrality gap: Dirichlet convex sets, sets with full-dimensional recession cones and sets that can be approximated by polyhedral sets. In the latter two cases, we provide estimates for the value of the integrality gap. Finally, we study the possibility of estimating the integrality gap of nonlinear convex mixed-integer programs via rational polyhedral approximations of their feasible regions and argue that, in general, such an approach may yield arbitrarily worse bounds compared to integrality gap estimations specifically derived by studying the associated nonlinear set.

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arXiv2025

Geoffrion's theorem beyond finiteness and rationality

Geoffrion's theorem is a fundamental result from mathematical programming assessing the quality of Lagrangian relaxation, a standard technique to get bounds for integer programs. An often implicit condition is that the set of feasible solutions is finite or described by rational linear constraints. However, we show through concrete examples that the conclusion of Geoffrion's theorem does not necessarily hold when this condition is dropped. We then provide sufficient conditions ensuring the validity of the result even when the feasible set is not finite and cannot be described using finitely-many linear constraints.

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arXiv2024

Generator Subadditive Functions for Mixed-Integer Programs

For equality-constrained linear mixed-integer programs (MIP) defined by rational data, it is known that the subadditive dual is a strong dual and that there exists an optimal solution of a particular form, termed generator subadditive function. Motivated by these results, we explore the connection between Lagrangian duality, subadditive duality and generator subadditive functions for general equality-constrained MIPs where the vector of variables is constrained to be in a monoid. We show that strong duality holds via generator subadditive functions under certain conditions. For the case when the monoid is defined by the set of all mixed-integer points contained in a convex cone, we show that strong duality holds under milder conditions and over a more restrictive set of dual functions. Finally, we provide some examples of applications of our results.

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