Full-dimensional Polytope Research Articles

Upgrading edges in the Graphical TSP

In this paper we present a generalization of the Graphical Traveling Salesman Problem (GTSP). Given a communication graph in which not all direct connections are necessarily possible, the Graphical Traveling Salesman Problem consists of finding the shortest tour that visits each node at least once. In this work, we assume the availability of a budget that allows to upgrade, i.e. reduce their traversal cost, some of the current connections and we propose the problem of designing the minimum cost tour using this budget. We propose and study a formulation for the problem, verifying that the polyhedron associated with the set of feasible solutions of a relaxed version of the problem is a full-dimensional polytope. We present families of valid inequalities that reinforce the model and pre-processing techniques to reduce the number of variables of the formulation. To solve the problem, we propose a branch-and-cut algorithm that uses the introduced valid inequalities, as well as a heuristic to obtain good upper bounds and a tailor-made branching strategy. Comprehensive computational experiments on a new set of benchmark instances are presented to assess the performance of this exact method.

Clean tangled clutters, simplices, and projective geometries

A clutter is clean if it has no delta or the blocker of an extended odd hole minor, and it is tangled if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in progress towards several open problems on ideal clutters, including the τ=2 Conjecture.Let C be a clean tangled clutter. It was recently proved that C has a fractional packing of value two. Collecting the supports of all such fractional packings, we obtain what is called the core of C. The core is a duplication of the cuboid of a set of 0−1 points, called the setcore of C.In this paper, we prove three results about the setcore. First, the convex hull of the setcore is a full-dimensional polytope containing the center point of the hypercube in its interior. Secondly, this polytope is a simplex if, and only if, the setcore is the cocycle space of a projective geometry over the two-element field. Finally, if this polytope is a simplex of dimension more than three, then C has the clutter of the lines of the Fano plane as a minor.Our results expose a fascinating interplay between the combinatorics and the geometry of clean tangled clutters.

The polytope algebra of generalized permutahedra

The polytope subalgebra of deformations of a zonotope can be endowed with the structure of a module over the Tits algebra of the corresponding hyperplane arrangement. We explore this construction and find relations between statistics on (signed) permutations and the module structure in the case of (type B) generalized permutahedra. In type B, the module structure surprisingly reveals that any family of generators (via signed Minkowski sums) for generalized permutahedra of type B will contain at least $2^{d-1}$ full-dimensional polytopes. We find a generating family of simplices attaining this minimum. Finally, we prove that the relations defining the polytope algebra are compatible with the Hopf monoid structure of generalized permutahedra.

Identifying a polytope by its fibre polytopes

It is shown that a full-dimensional polytope P is uniquely determined by its r-dimensional fibre polytopes when r ge 2. Further, if r ge 4 and the r-dimensional fibre polytopes are zonotopes, then P itself must be a zonotope.

A study of the quadratic semi-assignment polytope

We study a polytope which arises from a mixed integer programming formulation of the quadratic semi-assignment problem. We introduce an isomorphic projection and transform the polytope to a tractable full-dimensional polytope. As a result, some basic polyhedral properties, such as the dimension, the affine hull, and the trivial facets, are obtained. Further, we present valid inequalities called cut- and clique-inequalities and give complete characterizations for them to be facet-defining. We also discuss a simultaneous lifting of the clique-type facets. Finally, we show an application of the quadratic semi-assignment problem to hub location problems with some computational experiences.

On the general routing polytope

The General Routing Problem ( GRP) consists of finding a minimum length closed walk in an edge-weighted undirected graph, subject to containing certain sets of required nodes and edges. It is related to the Rural Postman Problem and the Graphical Traveling Salesman Problem. We examine the 0/1-polytope associated with the GRP introduced by Ghiani and Laporte [A branch-and-cut algorithm for the Undirected Rural Postman Problem, Math. Program. Ser. A 87 (3) (2000) 467–481]. We show that whenever it is not full-dimensional, the set of equations and facets can be characterized, and the polytope is isomorphic to the full-dimensional polytope associated with another GRP instance which can be obtained in polynomial time. We also offer a node-lifting method. Both results are applied to prove the facet-defining property of some classes of valid inequalities. As a tool, we study more general polyhedra associated to the GRP.

CONTROL OF DISCRETE-TIME PIECEWISE AFFINE SYSTEMS

In this paper we propose a new approach for the control of discrete-time piecewise affine hybrid systems on full-dimensional polytopes. The new strategy is divided into a local and a supervisory control problem. The local problem is to reach and cross one facet of a polytope ensuring that the next sample of the time-discrete trajectory is picked up in the adjacent polytope. Therefore, a local piecewise affine control law is obtained by solving an optimization problem, minimizing the retention period of the trajectory in the polytope or the quadratic sum of the input signal. This procedure is based on system inherent bounds. The supervisory control problem is to find a suitable combination of polytopes and local control strategies that transfers the trajectory to the operating point and keeps it there.

A control problem for affine dynamical systems on a full-dimensional polytope

Given an affine system on a full-dimensional polytope, the problem of reaching a particular facet of the polytope, using continuous piecewise-affine state feedback is studied. Necessary conditions and sufficient conditions for the existence of a solution are derived in terms of linear inequalities on the input vectors at the vertices of the polytope. Special attention is paid to affine systems on full-dimensional simplices. In this case, the necessary and sufficient conditions are equivalent and a constructive procedure yields an affine feedback control law, that solves the reachability problem under consideration.

On Numerical Solution of the Maximum Volume Ellipsoid Problem

In this paper we study practical solution methods for finding the maximum volume ellipsoid inscribing a given full-dimensional polytope in $\Re^n$ defined by a finite set of linear inequalities. Our goal is to design a general-purpose algorithmic framework that is reliable and efficient in practice. To evaluate the merit of a practical algorithm, we consider two key factors: the computational cost per iteration and the typical number of iterations required for convergence. In addition, numerical stability is an important factor. We investigate some new formulations upon which we build primal-dual type interior-point algorithms, and we provide theoretical justifications for the proposed formulations and algorithmic framework. Extensive numerical experiments have shown that one of the new algorithms is the method of choice among those tested.

Analytic centers and repelling inequalities

The new concepts of repelling inequalities, repelling paths, and prime analytic centers are introduced. A repelling path is a generalization of the analytic central path for linear programming, and we show that this path has a unique limit. Furthermore, this limit is the prime analytic center if the set of repelling inequalities contains only those constraints that “shape” the polytope. Because we allow lower dimensional polytopes, the proof techniques are non-standard and follow from data perturbation analysis. This analysis overcomes the difficulty that analytic centers of lower dimensional polytopes are not necessarily continuous with respect to the polytope's data representation. A second concept introduced here is that of the “prime analytic center”, in which we establish its uniqueness in the absence of redundant inequalities. Again, this is well known for full dimensional polytopes, but it is not immediate for lower dimensional polytopes because there are many different data representations of the same polytope, each without any redundant inequalities. These two concepts combine when we introduce ways in which repelling inequalities can interact.

Karmarkar's linear programming algorithm and Newton's method

This paper describes a full-dimensional version of Karmarkar's linear programming algorithm, theprojective scaling algorithm, which is defined for any linear program in ℝ n having a bounded, full-dimensional polytope of feasible solutions. If such a linear program hasm inequality constraints, then it is equivalent under an injective affine mappingJ:ℝ n →ℝ m to Karmarkar's original algorithm for a linear program in ℝ m havingm—n equality constraints andm inequality constraints. Karmarkar's original algorithm minimizes a potential functiong(x), and the projective scaling algorithm is equivalent to that version of Karmarkar's algorithm whose step size minimizes the potential function in the step direction. The projective scaling algorithm is shown to be a global Newton method for minimizing a logarithmic barrier function in a suitable coordinate system. The new coordinate system is obtained from the original coordinate system by a fixed projective transformationy = Ф(x) which maps the hyperplaneH opt ={x:〈c, x〉 =c 0} specified by the optimal value of the objective function to the “hyperplane at infinity”. The feasible solution set is mapped underФ to anunbounded polytope. Letf LB(y) denote the logarithmic barrier function associated to them inequality constraints in the new coordinate system. It coincides up to an additive constant with Karmarkar's potential function in the new coordinate system. Theglobal Newton method iterate y * for a strictly convex functionf(y) defined on a suitable convex domain is that pointy * that minimizesf(y) on the search ray {y+λ v N(y): λ≧0} wherev N(y) =−(∇2 f(y))−1(∇f(y)) is the Newton's method vector. If {x (k)} are a set of projective scaling algorithm iterates in the original coordinate system andy (k) =Ф(x (k)) then {y (k)} are a set of global Newton method iterates forf LB(y) and conversely. Karmarkar's algorithm with step size chosen to minimize the potential function is known to converge at least at a linear rate. It is shown (by example) that this algorithm does not have a superlinear convergence rate.