Related papers: Unique lifting of integer variables in minimal ine…
We explore the lifting question in the context of cut-generating functions. Most of the prior literature on this question focuses on cut-generating functions that have the unique lifting property. We develop a general theory for…
The goal of this paper is to derive new classes of valid convex inequalities for quadratically constrained quadratic programs (QCQPs) through the technique of lifting. Our first main result shows that, for sets described by one bipartite…
For a minimal inequality derived from a maximal lattice-free simplicial polytope in $\R^n$, we investigate the region where minimal liftings are uniquely defined, and we characterize when this region covers $\R^n$. We then use this…
We study the uniqueness of minimal liftings of cut-generating functions obtained from maximal lattice-free polyhedra. We prove a basic invariance property of unique minimal liftings for general maximal lattice-free polyhedra. This…
We investigate the properties of linear primitive liftings $\rho\colon \mathcal{L}^p(\mu)\to \mathcal{L}^p(\mu)$ for probability spaces $(X,\Sigma,\mu)$, which are linear maps selecting a representative from each class for almost everywhere…
The great advances of learning-based approaches in image processing and computer vision are largely based on deeply nested networks that compose linear transfer functions with suitable non-linearities. Interestingly, the most frequently…
In this paper, we investigate the mixed-integer nonlinear set with box constraints $X = \{(w,x)\in R\times Z^n:w\leq f(a^Tx),0\leq x\leq \mu\}$, where $f$ is a univariate concave function, $a\in R^n$, and $\mu\in Z^n_{++}$. This set arises…
Lifted inference reduces the complexity of inference in relational probabilistic models by identifying groups of constants (or atoms) which behave symmetric to each other. A number of techniques have been proposed in the literature for…
Lifting is a crucial technique in mixed integer programming (MIP) for generating strong valid inequalities, which serve as cutting planes to improve the branch-and-cut algorithm. We first propose an exact sequential lifting algorithm for…
Given a nonlinear, univariate, bounded, and differentiable function $f(x)$, this article develops a sequence of Mixed Integer Linear Programming (MILP) and Linear Programming (LP) relaxations that converge to the graph of $f(x)$ and its…
We propose a new method for separating valid inequalities for the epigraph of a function of binary variables. The proposed inequalities are disjunctive cuts defined by disjunctive terms obtained by enumerating a subset $I$ of the binary…
Functional lifting methods provide a tool for approximating solutions of difficult non-convex problems by embedding them into a larger space. In this work, we investigate a mathematically rigorous formulation based on embedding into the…
Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer…
Lifting methods allow to transform hard variational problems such as segmentation and optical flow estimation into convex problems in a suitable higher-dimensional space. The lifted models can then be efficiently solved to a global optimum,…
Recently, we proposed a class of inequalities called lifted bilinear cover inequalities, which are second-order cone representable convex inequalities, and are valid for a set described by a separable bilinear constraint together with…
How can one lift a functional defined on maps from a space X to a space Y into a functional defined on maps from X into P(Y) the space of probability distributions over Y? Looking at measure-valued maps can be interpreted as knowing a…
Despite the success of branch-and-cut methods for solving mixed integer bilevel linear optimization problems (MIBLPs) in practice, there are still gaps in both the theory and practice surrounding these methods. In the first part of this…
Lifted probabilistic inference algorithms exploit regularities in the structure of graphical models to perform inference more efficiently. More specifically, they identify groups of interchangeable variables and perform inference once per…
In this work we explore error-correcting codes derived from the "lifting" of "affine-invariant" codes. Affine-invariant codes are simply linear codes whose coordinates are a vector space over a field and which are invariant under…
Mixed integer linear programming (MILP) has seen a sharp rise in use for engineering optimization applications in recent years. Even for initially non-linear problems, it is often the method of choice. Then, the non-linear functions have to…