Polytopes

Grünbaum, Branko

doi:10.1007/978-1-4613-0019-9_3

Cited by 403 publications

(670 citation statements)

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“…This is illustrated in Figure 1, where (a) pictures an intersection cut derived from a vertexx = v 1 of P , while (b) and (c) show cuts derived from points v 2 and v 3 corresponding to infeasible basic solutions. Such points can be viewed as vertices (0-dimensional faces) of the hyperplane arrangement A(P ) in R n associated with P , which is the collection of hyperplanes defining the constraints of P (see, for instance, chapter 18 of [12]). A vertex of A(P ) is the intersection point of n hyperplanes of A(P ), i.e.…”

Section: Introductionmentioning

confidence: 99%

Generalized intersection cuts and a new cut generating paradigm

Balas

Margot

2011

Math. Program.

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Intersection cuts are generated from a polyhedral cone and a convex set S whose interior contains no feasible integer point. We generalize these cuts by replacing the cone with a more general polyhedron C. The resulting generalized intersection cuts dominate the original ones. This leads to a new cutting plane paradigm under which one generates and stores the intersection points of the extreme rays of C with the boundary of S rather than the cuts themselves. These intersection points can then be used to generate in a non-recursive fashion cuts that would require several recursive applications of some standard cut generating routine. A procedure is also given for strengthening the coefficients of the integer-constrained variables of a generalized intersection cut. The new cutting plane paradigm yields a new characterization of the closure of intersection cuts and their strengthened variants. This characterization is minimal in the sense that every one of the inequalities it uses defines a facet of the closure.

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Section: Introductionmentioning

confidence: 99%

Generalized intersection cuts and a new cut generating paradigm

Balas

Margot

2011

Math. Program.

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“…This inclusion is contradictory to the definition of a vertex of a convex set, see [21]. In other words, all elements of…”

Section: Assumption 53mentioning

confidence: 94%

“…The definition of a cell complex was presented by Grünbaum in [21]. For simplicity, a cell complex, in this paper, should be understood as a polyhedral partition whose face-to-face property is fulfilled, i.e., for any pair of regions, the intersection of faces is either empty or a common face.…”

Section: Notation and Definitionsmentioning

confidence: 99%

Constructive Solution of Inverse Parametric Linear/Quadratic Programming Problems

Nguyễn

Olaru

Rodríguez-Ayerbe

et al. 2016

J Optim Theory Appl

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Parametric convex programming has received a lot of attention, since it has many applications in chemical engineering, control engineering, signal processing, etc. Further, inverse optimality plays an important role in many contexts, e.g., image processing, motion planning. This paper introduces a constructive solution of the inverse optimality problem for the class of continuous piecewise affine functions. The main idea is based on the convex lifting concept. Accordingly, an algorithm to construct convex liftings of a given convexly liftable partition will be put forward. Following this idea, an important result will be presented in this article: any continuous piecewise affine function defined over a polytopic partition is the solution of a parametric linear/quadratic programming problem. Regarding linear optimal control, it will be shown that any continuous piecewise affine control law can be obtained via a linear optimal control problem with the control horizon at most equal to 2 prediction steps.

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“…In this case, the faces of a plane realization are uniquely determined up to the choice of the outer face. By Steinitz theorem (see Grünbaum [6]), any polyhedral graph can be realized as a convex polyhedron. A polyhedral graph is inscribable if its corresponding convex polyhedron is combinatorially equivalent to the edges and vertices of the convex hull of a set of noncoplanar points on the surface of the sphere.…”

Section: Preliminariesmentioning

confidence: 99%