Equivariant perturbation in Gomory and Johnson's infinite group problem (V). Software for the continuous and discontinuous 1-row case

Abstract

We present software for investigations with cut-generating functions in the Gomory-Johnson model and extensions, implemented in the computer algebra system SageMath.

Equivariant perturbation in Gomory and Johnson’s infinite group problem. VI. The curious case of two-sided discontinuous minimal valid functions

“…Note that ∆π is affine over F , so the condition is equivalent to ∆π(u, v) = 0 for any (u, v) ∈ vert(F ). When π is discontinuous, following [14,15], we say that a face F ∈ ∆P is additive if F is contained in a face F ∈ ∆P such that ∆π F (x, y) = 0 for any (x, y) ∈ F .…”

“…Note that all of the above-mentioned families of extreme functions are one-sided continuous at the origin, either from the left or from the right. To explain the significance of this observation, we will outline the structure of an extremality proof, using the notion of effective perturbations introduced in [14,15]. Let π : R → R + be a minimal valid function.…”

Characterization and Approximation of Strong General Dual Feasible Functions

“…We will use the term "piecewise linear" throughout the paper without explanation. We refer readers to [13] for precise definitions of "piecewise linear" functions in both continuous and discontinuous cases. Although piecewise linearity is not implied in the definition of gDFF, nearly all known gDFFs are piecewise linear.…”

On the Notions of Facets, Weak Facets, and Extreme Functions of the Gomory–Johnson Infinite Group Problem

“…We remark that there is no inclusion relation between the limit-additivities captured in the set families E • (ψ, P) and E • (ψ , P), as illustrated in the diagrams in Figure 4. We will explain these diagrams on an example only; see [18], where these types of diagrams for discontinuous piecewise linear functions were introduced, for a full discussion. 4 Consider ( 3 8 , 3 8 ) as a vertex of the face F ∈ ∆P that is the triangle to the northeast of it.…”

“…This function has 40 breakpoints 0 = x 0 < x 1 < · · · < x 39 < x 40 = 1 within [0, 1]. It has two special intervals (l, u) = (x 17 , x 18 ) and (f −u, f −l) = (x 19 , x 20 ), where f = x 37 = 4 5 , l = 219 800 , u = 269 800 , on which every nonzero perturbation is microperiodic, namely invariant under the action of the dense additive group T = t 1 , t 2 Z , where t 1 = a 1 − a 0 = x 10 − x 6 = 77 7752 √ 2 and t 2 = a 2 − a 0 = x 13 − x 6 = 77 2584 . Below we prove that it furnishes another separation.…”