Polyhedral Computations Research Articles

The maximum 2D subarray polytope: Facet-inducing inequalities and polyhedral computations

Given a matrix with real-valued entries, the maximum 2D subarray problem consists in finding a rectangular submatrix with consecutive rows and columns maximizing the sum of its entries. In this work we start a polyhedral study of an integer programming formulation for this problem. We thus define the 2D subarray polytope , explore conditions ensuring the validity of linear inequalities, and provide several families of facet-inducing inequalities. We also report computational experiments assessing the reduction of the dual bound for the linear relaxation achieved by these families of inequalities.

A task‐based approach to parallel parametric linear programming solving, and application to polyhedral computations

SummaryParametric linear programming is a central operation for polyhedral computations, as well as in certain control applications. Here, we propose a task‐based scheme for parallelizing it, with quasi‐linear speedup over large problems. This type of parallel applications is challenging, because several tasks might be computing the same region. In this article, we are presenting the algorithm itself with a parallel redundancy elimination algorithm, and conducting a thorough performance analysis.

Geometry and algorithms for upper triangular tropical matrix identities

We provide geometric methods and algorithms to verify, construct and enumerate pairs of words (of specified length over a fixed m-letter alphabet) that form identities in the semigroup UTn(T) of n×n upper triangular tropical matrices. In the case n=2 these identities are precisely those satisfied by the bicyclic monoid, whilst in the case n=3 they form a subset of the identities which hold in the plactic monoid of rank 3. To each word we associate a signature sequence of lattice polytopes, and show that two words form an identity for UTn(T) if and only if their signatures are equal. Our algorithms are thus based on polyhedral computations and achieve optimal time complexity in some cases. For n=m=2 we prove a Structural Theorem, which allows us to quickly enumerate the pairs of words of fixed length which form identities for UT2(T). This allows us to recover a short proof of Adjan's theorem on minimal length identities for the bicyclic monoid, and to construct minimal length identities for UT3(T), providing counterexamples to a conjecture of Izhakian in this case. We conclude with six conjectures at the intersection of semigroup theory, probability and combinatorics, obtained through analysing the outputs of our algorithms.

Structure and Interpretation of Dual-Feasible Functions

We study two techniques to obtain new families of classical and general Dual-Feasible Functions: a conversion from minimal Gomory–Johnson functions; and computer-based search using polyhedral computation and an automatic maximality and extremality test.

A Multiparametric Quadratic Programming Algorithm With Polyhedral Computations Based on Nonnegative Least Squares

Model predictive control (MPC) is one of the most successful techniques adopted in industry to control multivariable systems under constraints on input and output variables. To circumvent the main drawback of MPC, i.e., the need to solve a Quadratic Program (QP) on line to compute the control action, explicit MPC was proposed in the past to precompute the control law off line using multiparametric QP (mpQP). The resulting form of the MPC law is piecewise affine, which is extremely easy to code, can be computed online by simple arithmetic operations, and requires a maximum number of iterations that can be exactly determined a priori. On the other hand, the offline computations to solve the mpQP problem require detecting emptiness, full-dimensionality, and minimal hyperplane representations of polyhedra, and other computational geometric operations. While most of the existing methods solve such operations via linear programming, the approach proposed in this paper relies on a nonnegative least squares (NNLS) solver that is very simple to code and fast to execute. In addition, the new approach exploits QP duality to identify and construct critical regions and to handle degeneracy issues.

Computing symmetry groups of polyhedra

AbstractKnowing the symmetries of a polyhedron can be very useful for the analysis of its structure as well as for practical polyhedral computations. In this note, we study symmetry groups preserving the linear, projective and combinatorial structure of a polyhedron. In each case we give algorithmic methods to compute the corresponding group and discuss some practical experiences. For practical purposes the linear symmetry group is the most important, as its computation can be directly translated into a graph automorphism problem. We indicate how to compute integral subgroups of the linear symmetry group that are used, for instance, in integer linear programming.

A-tint: A polymake extension for algorithmic tropical intersection theory

In this paper we study algorithmic aspects of tropical intersection theory. We analyze how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus of this paper is the study of moduli spaces, where the underlying combinatorics of the varieties involved allow a much more efficient way of computing certain tropical cycles. The algorithms discussed here have been implemented in an extension for polymake, a software for polyhedral computations.

Supervisory control of Petri nets using polyhedral regions

We propose to use polyhedral computations to check the reachability and co-reachability of Petri nets, and design supervisors enforcing polyhedral invariance. To this aim, we first state a semi-algebraic behavioral model for Petri nets, derived from the marking equation. From it, we derive explicit characterizations of the sets of k-reachable marking vectors, in terms of polyhedral regions of a real vector space. This permits to calculate the region of reachable marking vectors, when the considered Petri net is bounded, as the projection of an integral polyhedron. The polyhedron is computed using the Γ-algorithm, in polynomial time. We propose an algorithm to this computation. We adapt the method to further compute the polytope of reachable, co-reachable and safe marking vectors, meeting a given set of polyhedral constraints, and finally apply it to design a supervising controller. We illustrate the method on the supervisor design of a simple AGV system.

Data locality optimization of interference graphs based on polyhedral computations

In achieving high performance on modern architectures it is critical to make effective use of the memory hierarchy. There are compiler-directed locality enhancement techniques that allow the transformation of program to achieve a higher locality: loop transformations, which are constrained by data dependences and data layout transformations, which have a global impact on the program locality. Due to these drawbacks, there must be a unification of the two techniques to achieve the benefits of both. In this paper, a novel unification of these techniques is presented. Using a model based on parameterized polyhedra and introducing new concepts, we propose a data locality optimization algorithm.

Exact join detection for convex polyhedra and other numerical abstractions

Deciding whether the union of two convex polyhedra is itself a convex polyhedron is a basic problem in polyhedral computations; having important applications in the field of constrained control and in the synthesis, analysis, verification and optimization of hardware and software systems. In such application fields though, general convex polyhedra are just one among many, so-called, numerical abstractions, which range from restricted families of (not necessarily closed) convex polyhedra to non-convex geometrical objects. We thus tackle the problem from an abstract point of view: for a wide range of numerical abstractions that can be modeled as bounded join-semilattices—that is, partial orders where any finite set of elements has a least upper bound—we show necessary and sufficient conditions for the equivalence between the lattice-theoretic join and the set-theoretic union. For the case of closed convex polyhedra—which, as far as we know, is the only one already studied in the literature—we improve upon the state-of-the-art by providing a new algorithm with a better worst-case complexity. The results and algorithms presented for the other numerical abstractions are new to this paper. All the algorithms have been implemented, experimentally validated, and made available in the Parma Polyhedra Library.

Applications of polyhedral computations to the analysis and verification of hardware and software systems

Convex polyhedra are the basis for several abstractions used in static analysis and computer-aided verification of complex and sometimes mission-critical systems. For such applications, the identification of an appropriate complexity–precision trade-off is a particularly acute problem, so that the availability of a wide spectrum of alternative solutions is mandatory. We survey the range of applications of polyhedral computations in this area; give an overview of the different classes of polyhedra that may be adopted; outline the main polyhedral operations required by automatic analyzers and verifiers; and look at some possible combinations of polyhedra with other numerical abstractions that have the potential to improve the precision of the analysis. Areas where further theoretical investigations can result in important contributions are highlighted.

PHAVer: algorithmic verification of hybrid systems past HyTech

In 1995, HyTech broke new ground as a potentially powerful tool for verifying hybrid systems. But due to practical and systematic limitations it is only applicable to relatively simple systems. We address the main problems of HyTech with PHAVer, a new tool for the exact verification of safety properties of hybrid systems with piecewise constant bounds on the derivatives, so-called linear hybrid automata. Affine dynamics are handled by on-the-fly overapproximation and partitioning of the state space based on user-provided constraints and the dynamics of the system. PHAVer features exact arithmetic in a robust implementation that, based on the Parma Polyhedra Library, supports arbitrarily large numbers. To force termination and manage the complexity of the polyhedral computations, we propose methods to conservatively limit the number of bits and constraints of polyhedra. Experimental results for a navigation benchmark and a tunnel diode circuit demonstrate the effectiveness of the approach.

Classification of eight-dimensional perfect forms

In this paper, we classify the perfect lattices in dimension88. There are1091610916of them. Our classification heavily relies on exploiting symmetry in polyhedral computations. Here we describe algorithms making the classification possible.

Predictive control for hybrid systems. Implications of polyhedral pre-computations

The present paper deals with the predictive control laws for hybrid systems. The modelling formalism used will be the Mixed Logical Dynamical (MLD) which offers the advantage of a compact expression of the dynamics in terms of linear equalities and inequalities on the logical and continuous-time states and inputs. Being an optimization-based control technique, the predictive control needs an efficient implementation scheme in order to be effective in real time. Several studies assess the importance of the prediction horizon and the terminal constraints due to their implications in the structure of the associated optimal control problem. Lately it has been shown that as long as the constraints remain linear, the polyhedral computations can serve as tools for the migration of the on-line computational effort to off-line explicit constructions in terms of explicit solutions which can avoid a costly on-line optimum seeking and thus pushing the application of predictive laws to even higher sampling rates. This paper reviews the on-line optimization techniques proposed for the predictive control of hybrid systems based on mixed integer optimization problems. Further, the explicit solutions are analyzed using a parameterized polyhedron approach.

From the zonotope construction to the Minkowski addition of convex polytopes

A zonotope is the Minkowski addition of line segments in R d . The zonotope construction problem is to list all extreme points of a zonotope given by its line segments. By duality, it is equivalent to the arrangement construction problem—that is, to generate all regions of an arrangement of hyperplanes. By replacing line segments with convex V-polytopes, we obtain a natural generalization of the zonotope construction problem: the construction of the Minkowski addition of k polytopes. Gritzmann and Sturmfels studied this general problem in various aspects and presented polynomial algorithms for the problem when one of the parameters k or d is fixed. The main objective of the present work is to introduce an efficient algorithm for variable d and k. Here we call an algorithm efficient or polynomial if it runs in time bounded by a polynomial function of both the input size and the output size. The algorithm is a natural extension of a known algorithm for the zonotope construction, based on linear programming and reverse search. It is compact, highly parallelizable and very easy to implement. This work has been motivated by the use of polyhedral computation for optimal tolerance determination in mechanical engineering.

Computation of elementary modes: a unifying framework and the new binary approach

BackgroundMetabolic pathway analysis has been recognized as a central approach to the structural analysis of metabolic networks. The concept of elementary (flux) modes provides a rigorous formalism to describe and assess pathways and has proven to be valuable for many applications. However, computing elementary modes is a hard computational task. In recent years we assisted in a multiplication of algorithms dedicated to it. We require a summarizing point of view and a continued improvement of the current methods.ResultsWe show that computing the set of elementary modes is equivalent to computing the set of extreme rays of a convex cone. This standard mathematical representation provides a unified framework that encompasses the most prominent algorithmic methods that compute elementary modes and allows a clear comparison between them. Taking lessons from this benchmark, we here introduce a new method, the binary approach, which computes the elementary modes as binary patterns of participating reactions from which the respective stoichiometric coefficients can be computed in a post-processing step. We implemented the binary approach in FluxAnalyzer 5.1, a software that is free for academics. The binary approach decreases the memory demand up to 96% without loss of speed giving the most efficient method available for computing elementary modes to date.ConclusionsThe equivalence between elementary modes and extreme ray computations offers opportunities for employing tools from polyhedral computation for metabolic pathway analysis. The new binary approach introduced herein was derived from this general theoretical framework and facilitates the computation of elementary modes in considerably larger networks.

Convexity recognition of the union of polyhedra

In this paper we consider the following basic problem in polyhedral computation: Given two polyhedra in R d , P and Q, decide whether their union is convex, and, if so, compute it. We consider the three natural specializations of the problem: (1) when the polyhedra are given by halfspaces (H-polyhedra), (2) when they are given by vertices and extreme rays (V-polyhedra), and (3) when both H- and V-polyhedral representations are available. Both the bounded (polytopes) and the unbounded case are considered. We show that the first two problems are polynomially solvable, and that the third problem is strongly-polynomially solvable.

The edge-weighted clique problem: Valid inequalities, facets and polyhedral computations

Let K n =( V, E) be the complete undirected graph with weights c e associated to the edges in E. We consider the problem of finding the subclique C=( U, F) of K n such that the sum of the weights of the edges in F is maximized and | U|⩽ b, for some b∈[1,…, n]. This problem is called the Maximum Edge-Weighted Clique Problem (MEWCP) and is NP-hard. In this paper we investigate the facial structure of the polytope associated to the MEWCP introducing new classes of facet defining inequalities. Computational experiments with a branch-and-cut algorithm are reported confirming the strength of these inequalities. All instances with up to 48 nodes could be solved without entering into the branching phase. Moreover, we show that some of these new inequalities also define facets of the Boolean Quadric Polytope and generalize many of the previously known inequalities for this well-studied polytope.