Related papers: Equivariant Perturbation in Gomory and Johnson's I…
In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory-Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We…
We give an algorithm for testing the extremality of minimal valid functions for Gomory and Johnson's infinite group problem that are piecewise linear (possibly discontinuous) with rational breakpoints. This is the first set of necessary and…
We give an algorithm for testing the extremality of a large class of minimal valid functions for the two-dimensional infinite group problem.
We present software for investigations with cut-generating functions in the Gomory--Johnson model and extensions, implemented in the computer algebra system SageMath.
We give a variant of Basu-Hildebrand-Molinaro's approximation theorem for continuous minimal valid functions for Gomory-Johnson's infinite group problem by piecewise linear two-slope extreme functions [Minimal cut-generating functions are…
We study an abstract setting for cutting planes for integer programming called the infinite group problem. In this abstraction, cutting planes are computed via cut generating function that act on the simplex tableau. In this function space,…
In this note we announce the availability of an electronic compendium of extreme functions for Gomory--Johnson's infinite group problem. These functions serve as the strongest cut-generating functions for integer linear optimization…
We consider Gomory and Johnson's infinite group model with a single row. Valid inequalities for this model are expressed by valid functions and it has been recently shown that any valid function is dominated by some nonnegative valid…
We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory-Johnson model. These notions were known to coincide for continuous piecewise linear functions…
The paper is devoted to a comprehensive second-order study of a remarkable class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to…
We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory-Johnson model. These notions were known to coincide for continuous piecewise linear functions…
For the one dimensional infinite group relaxation, we construct a sequence of extreme valid functions that are piecewise linear and such that for every natural number $k\geq 2$, there is a function in the sequence with $k$ slopes. This…
This paper is devoted to second-order variational analysis of a rather broad class of extended-real-valued piecewise liner functions and their applications to various issues of optimization and stability. Based on our recent explicit…
In Mathematical Programming 2003, Gomory and Johnson conjecture that the facets of the infinite group problem are always generated by piecewise linear functions. In this paper we give an example showing that the Gomory-Johnson conjecture is…
In a recent paper, Basu, Hildebrand, and Molinaro established that the set of continuous minimal functions for the 1-dimensional Gomory-Johnson infinite group relaxation possesses a dense subset of extreme functions. The $n$-dimensional…
In this article we introduce a class of discontinuous almost automorphic functions which appears naturally in the study of almost automorphic solutions of differential equations with piecewise constant argument. Their fundamental properties…
We prove that any minimal valid function for the k-dimensional infinite group relaxation that is piecewise linear with at most k+1 slopes and does not factor through a linear map with non-trivial kernel is extreme. This generalizes a…
We propose and analyze a discontinuous least squares finite element method for solving the indefinite time-harmonic Maxwell equations. The scheme is based on the $L^2$ norm least squares functional with the weak imposition of the continuity…
This paper aims to provide various applications for second-order variational analysis of extended-real-valued piecewise liner functions recently obtained in [1]. We mainly focus here on establishing relationships between full stability of…
There are many significant applied contexts that require the solution of discontinuous optimization problems in finite dimensions. Yet these problems are very difficult, both computationally and analytically. With the functions being…