Submodular set functions play an important role in integer programming and combinatorial optimization.
Increasingly, applications in resource allocation and machine learning have led to generalized submodular
functions defined over general integer or continuous domains. In our study, we focus on the notion of
Diminishing Returns (DR) submodularity and the problem of minimizing DR-submodular functions over
mixed-integer feasible sets defined by box constraints and monotonicity constraints. We propose multiple
classes of valid inequalities and further show that these inequalities, along with trivial constraints, fully
describe the convex hull of the epigraph of DR-submodular functions under the aforementioned constraints.
Our results hold in the original decision space, which avoids pseudo-polynomial expansions that are used to
handle general integer variables in the existing literature.

We study linear bilevel programming problems whose lower-level objective is given by a random cost vector with known distribution. We consider the case where this distribution is nonatomic, allowing to reformulate the problem of the leader using the Bayesian approach in the sense of Salas and Svensson (2023) with a vertex-supported belief. We prove that this formulation is piecewise affine over the so-called chamber complex of the feasible set of the high point relaxation. We propose two algorithmic approaches to solve general problems enjoying this last property. The first one is based on enumerating the vertices of the chamber complex. The second one is a Monte-Carlo approximation scheme based on the fact that randomly drawn points of the domain lie, with probability 1, in the interior of full-dimensional chambers, where the problem (restricted to this chamber) can be reduced to a linear program. Finally, we evaluate these methods through computational experiments showing both approaches' advantages and challenges.

The recently introduced continuous quadrant penalty formulation of logical disjunctive constraints constitutes a continuous-optimization alternative to the classical formulations (such as the bigM formulation) based on the introduction of binary variables.
We focus on nonlinear problems whose only combinatorial aspect comes from their logical constraints, and build on the continuous quadrant penalty, that yields to a continuous nonconvex problem, to derive an efficient computation of upper bounds to be used within Branch-and-Bound-based multi-step approaches.
We show on two problems, respectively from discrete geometry and arising in air traffic management optimization, that our approach is effective at speeding up the computational convergence.