Polyhedral Techniques Research Articles

On solving terminal approach and evasion problems for linear discrete-time systems under state constraints

The paper is devoted to elaboration of polyhedral techniques for solving two control problems for linear discrete-time systems with uncertainties under state constraints, namely, the terminal approach problem and the terminal evasion one. Such problems arise in systems with two controls, where the aim of the first is to steer the trajectory onto a given terminal set at a given instant without violating the state constraints, the aim of the other is opposite. It is assumed that the terminal set is a parallelepiped, the controls are bounded by parallelotope-valued constraints, and the state constraints are given in the form of so-called zones. We present techniques for solving both problems basing on polyhedral (parallelotope-valued or parallelepiped-valued) tubes. The techniques for solving the approach problem were proposed by the author earlier, but here additional properties of them are investigated. In particular, for the case without state constraints, guaranteed estimates are found for the trajectory that ensure that it is inside the tube. Convenient sufficient conditions are given to guarantee the obtaining of nondegenerate cross-sections during the calculations. For the evasion problem, a common solution scheme is considered, and then polyhedral techniques are proposed. The whole parametric families of external and internal polyhedral estimates for the solvability tubes for both problems are presented and compared. An illustrative example is given.

New Graph Algorithms via Polyhedral Techniques

New Graph Algorithms via Polyhedral Techniques

New graph algorithms via polyhedral techniques

Abstract This article gives a short overview of my dissertation, where new algorithms are given for two fundamental graph problems. We develop novel ways of using linear programming formulations, even exponential-sized ones, to extract structure from problem instances and to guide algorithms in making progress. The first part of the dissertation addresses a benchmark problem in combinatorial optimization: the asymmetric traveling salesman problem (ATSP). It consists in finding the shortest tour that visits all vertices of a given edge-weighted directed graph. A ρ-approximation algorithm for ATSP is one that runs in polynomial time and always produces a tour at most ρ times longer than the shortest tour. Finding such an algorithm with constant ρ had been a long-standing open problem. Here we give such an algorithm. The second part of the dissertation addresses the perfect matching problem. We have known since the 1980s that it has efficient parallel algorithms if the use of randomness is allowed. However, we do not know if randomness is necessary – that is, whether the matching problem is in the class NC. We show that it is in the class quasi-NC. That is, we give a deterministic parallel algorithm that runs in poly-logarithmic time on quasi-polynomially many processors.

An optimal monotone contention resolution scheme for bipartite matchings via a polyhedral viewpoint

Relaxation and rounding approaches became a standard and extremely versatile tool for constrained submodular function maximization. One of the most common rounding techniques in this context are contention resolution schemes. Such schemes round a fractional point by first rounding each coordinate independently, and then dropping some elements to reach a feasible set. Also the second step, where elements are dropped, is typically randomized. This leads to an additional source of randomization within the procedure, which can complicate the analysis. We suggest a different, polyhedral viewpoint to design contention resolution schemes, which avoids to deal explicitly with the randomization in the second step. This is achieved by focusing on the marginals of a dropping procedure. Apart from avoiding one source of randomization, our viewpoint allows for employing polyhedral techniques. Both can significantly simplify the construction and analysis of contention resolution schemes. We show how, through our framework, one can obtain an optimal monotone contention resolution scheme for bipartite matchings, which has a balancedness of 0.4762. So far, only very few results are known about optimality of monotone contention resolution schemes. Our contention resolution scheme for the bipartite case also improves the lower bound on the correlation gap for bipartite matchings. Furthermore, we derive a monotone contention resolution scheme for matchings that significantly improves over the previously best one. More precisely, we obtain a balancedness of 0.4326, improving on a prior 0.1997-balanced scheme. At the same time, our scheme implies that the currently best lower bound on the correlation gap for matchings is not tight. Our results lead to improved approximation factors for various constrained submodular function maximization problems over a combination of matching constraints with further constraints.

Tuning iteration space slicing based tiled multi-core code implementing Nussinov\u2019s RNA folding

BackgroundRNA folding is an ongoing compute-intensive task of bioinformatics. Parallelization and improving code locality for this kind of algorithms is one of the most relevant areas in computational biology. Fortunately, RNA secondary structure approaches, such as Nussinov’s recurrence, involve mathematical operations over affine control loops whose iteration space can be represented by the polyhedral model. This allows us to apply powerful polyhedral compilation techniques based on the transitive closure of dependence graphs to generate parallel tiled code implementing Nussinov’s RNA folding. Such techniques are within the iteration space slicing framework – the transitive dependences are applied to the statement instances of interest to produce valid tiles. The main problem at generating parallel tiled code is defining a proper tile size and tile dimension which impact parallelism degree and code locality.ResultsTo choose the best tile size and tile dimension, we first construct parallel parametric tiled code (parameters are variables defining tile size). With this purpose, we first generate two nonparametric tiled codes with different fixed tile sizes but with the same code structure and then derive a general affine model, which describes all integer factors available in expressions of those codes. Using this model and known integer factors present in the mentioned expressions (they define the left-hand side of the model), we find unknown integers in this model for each integer factor available in the same fixed tiled code position and replace in this code expressions, including integer factors, with those including parameters. Then we use this parallel parametric tiled code to implement the well-known tile size selection (TSS) technique, which allows us to discover in a given search space the best tile size and tile dimension maximizing target code performance.ConclusionsFor a given search space, the presented approach allows us to choose the best tile size and tile dimension in parallel tiled code implementing Nussinov’s RNA folding. Experimental results, received on modern Intel multi-core processors, demonstrate that this code outperforms known closely related implementations when the length of RNA strands is bigger than 2500.

State Estimates of Bilinear Discrete-Time Systems with Integral Constraints through Polyhedral Techniques

Abstract We consider bilinear discrete-time systems for the case when interval bounds on the coefficients of the system are imposed, additive input terms are restricted by integral constraints, and initial states are restricted by given sets. Several ways for constructing external parallelepiped-valued estimates of reachable sets of such systems are proposed. One of the above techniques is based on obtaining recurrence relations for reachable sets in the ”extended” phase space and constructing corresponding estimates in the form of polytopes of some special shape.

Robust Model Predictive Controller for Uncertain Piecewise Affine Systems

This paper considers discrete-time, uncertain piecewise affine systems affected by both polytopic parameter variations and bounded disturbances, where the uncertainty can be presented in a polytopes framework. A new Robust Model Predictive Control (RMPC) algorithm based on polyhedral techniques that checks the attainability of a state space region is presented. The developed approach computes in its first stage the maximal set in the state space from which states can be driven to the target region while taking into account the system’s uncertainties. Thereafter, a RMPC algorithm that guarantees to drive the system states from the maximal set to the target region is proposed. The proposed algorithm not only gives a simple and fast sub-optimal solution which considerably reduces the online computation, it guarantees the feasibility and the stability of the solution. The proposed algorithms are applied in simulation to a two-tank example.

The traveling salesman problem on cubic and subcubic graphs

We study the traveling salesman problem (TSP) on the metric completion of cubic and subcubic graphs, which is known to be NP-hard. The problem is of interest because of its relation to the famous 4/3-conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal value of a TSP instance and that of its linear programming relaxation (the subtour elimination relaxation), is 4/3. We present the first algorithm for cubic graphs with approximation ratio 4/3. The proof uses polyhedral techniques in a surprising way, which is of independent interest. In fact we prove constructively that for any cubic graph on TeX vertices a tour of length TeX exists, which also implies the 4/3-conjecture, as an upper bound, for this class of graph-TSP. Recently, Momke and Svensson presented an algorithm that gives a 1.461-approximation for graph-TSP on general graphs and as a side result a 4/3-approximation algorithm for this problem on subcubic graphs, also settling the 4/3-conjecture for this class of graph-TSP. The algorithm by Momke and Svensson is initially randomized but the authors remark that derandomization is trivial. We will present a different way to derandomize their algorithm which leads to a faster running time. All of the latter also works for multigraphs.

Some approaches for solving the general [formula omitted]-design existence problem and other related problems

In this short survey, some approaches that can be used to solve the generalized (t,k)-design problem are considered. Special cases of the generalized (t,k)-design problem include well-known conjectures for t-designs, degree sequences of graphs and hypergraphs, and partial Steiner systems. Also described are some related problems such as the characterization of f-vectors of pure simplicial complexes, which are well known but little understood. Some suggestions how enumerative and polyhedral techniques may help are also described.

Optimization of Output Feedback Control Under Set-Membership Uncertainty

This paper presents a description of solution approaches to the problem of output feedback control under unknown but bounded disturbances with hard bounds on the controls and the uncertain items. The problem is treated within a finite horizon which requires to track the system dynamics throughout the whole time interval rather than through asymptotic properties. The demand for such solutions is motivated by increasing number of applications. The suggested approaches are designed as a combination of Hamiltonian techniques in the form of generalized dynamic programming with those of set-valued analysis and problems on minimax. The paper indicates the crucial role of properly selecting the on-line generalized state of the system in the form of information states or information sets. The description ranges from theoretical schemes to computational routes with emphasis on the possibility of treating the overall problem through only finite-dimensional methods. The results apply to nonlinear systems, with more details for linear models. It turns out that in the last case, while moving through calculations, one may avoid the fairly difficult stage of integrating HJB equations. The procedures are here confined to ordinary differential equations and ellipsoidal or polyhedral techniques. The suggested schemes are also quite appropriate for parallel computation.

Combinatorial Optimization

Combinatorial Optimization remains a very lively discipline with strong connections to Combinatorics, Graph Theory, Geometry, and Integer Programming. For over thirty years, Oberwolfach workshops have had a central role in shaping the field, being the unique setting where the entire spectrum of the subject is covered, from fundamental theory to practical computation. The 2008 Combinatorial Optimization Workshop was again a great success. We would like to take this opportunity to thank all speakers for the great care they put into preparing their contributions, and to thank all participants for their active discussions that led to a truly stimulating meeting. Just like our predecessors, we are happy that we could attract many of the strongest theoreticians as well as the most successful practitioners to the workshop. Even though the team of organizers has dynamically changed over the last thirty years, a format has emerged that has been quite stable in the last decade, namely a mix of a few pre-arranged focus talks and short presentations of the participants that are scheduled during the conference. In the focus talks, selected experts survey the state-of-the-art in current hot topics and point out new mathematical challenges. While we have largely adopted this format, we have lowered the number of short presentations in favor of more time that can be spent for joint research in small groups, taking up a suggestion made by the director during the previous Combinatorial Optimization Workshop held in 2005. We have had very positive feedback from workshop participants that this measure enhances the conditions for new cooperation between participants that can be expected to lead to continued joint work and publications afterwards. Each day, the program started with a pre-arranged one hour focus presentation, followed by two thirty-minute presentations in the morning and three thirty-minute presentations before dinner. This year's focus talks featured topics that bring together techniques from other areas of mathematics to explore new approaches for attacking well-known combinatorial problems. Tibor Jord'an led off the workshop with a talk covering methods in network localization, making use of ideas from combinatorial rigidity. Tuesday's talk was by Kazuo Murota, discussing the new area of discrete convex analysis, where combinatorial analogues of basic convex results are being developed, mainly in the Japanese research community. Friedrich Eisenbrand presented new directions in the application of techniques from the geometry of numbers in Wednesday's talk, including several new open problems in the area of integer programming. On Thursday, Monique Laurent gave an in-depth treatment of the use of semidefinite programming techniques applied to coloring problems in graphs. Andr{'a}s Seb\H{o} gave the final focus talk on Friday, describing the application of polyhedral techniques to graph optimization problems. A new addition to this year's workshop was an evening session devoted to computational integer programming, featuring discussions led by Robert Bixby, Sanjeeb Dash, and Alexander Martin. The session highlights one of the remarkable achievements of the field in general and the Oberwolfach meetings in particular, namely, a very fast transition form basic theoretical breakthroughs to commercially successful software. Within the course of the three-year cycle of the Oberwolfach meetings, proposed new techniques work their way from academia to the leading mixed-integer programming software, delivered throughout the industrial sector. The broad mix of researchers at the Combinatorial Optimization Workshops directly contributes to this rapid transfer.

Sur la forme de la boule unité de la norme stable unidimensionnelle

We study the stable norm on the first homology of a Riemannian polyhedron. In the one-dimensional case (metric graphs), the geometry of the unit ball of this norm is completely described by the combinatorial structure of the graph. For a smooth manifold of dimension ≥3 and using polyhedral techniques, we show that a large class of polytopes appears as unit ball of the stable norm associated to some metric conformal to a given one.

Polyhedral techniques in combinatorial optimization II: applications and computations

The polyhedral approach is one of the most powerful techniques available for solving hard combinatorial optimization problems. The main idea behind the technique is to consider the linear relaxation of the integer combinatorial optimization problem, and try to iteratively strengthen the linear formulation by adding violated strong valid inequalities, i.e., inequalities that are violated by the current fractional solution but satisfied by all feasible solutions, and that define high‐dimensional faces, preferably facets, of the convex hull of feasible solutions. If we have the complete description of the convex hull of feasible solutions at hand all extreme points of this formulation are integral, which means that we can solve the problem as a linear programming problem. Linear programming problems are known to be computationally easy. In Part 1 of this article we discuss theoretical aspects of polyhedral techniques. Here we will mainly concentrate on the computational aspects. In particular we discuss how polyhedral results are used in cutting plane algorithms. We also consider a few theoretical issues not treated in Part 1, such as techniques for proving that a certain inequality is facet defining, and that a certain linear formulation gives a complete description of the convex hull of feasible solutions. We conclude the article by briefly mentioning some alternative techniques for solving combinatorial optimization problems.

Polyhedral techniques in combinatorial optimization I: Theory

Combinatorial optimization problems appear in many disciplines ranging from management and logistics to mathematics, physics, and chemistry. These problems are usually relatively easy to formulate mathematically, but most of them are computationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable progress in techniques based on the polyhedral description of combinatorial problems, leading to a large increase in the size of several problem types that can be solved. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. Ideally we can then solve the problem as a linear programming problem, which can be done efficiently. The purpose of this manuscript is to give an overview of the developments in polyhedral theory, starting with the pioneering work by Dantzig. Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also present some modern applications, and computational experience.

On the integral dicycle packings and covers and the linear ordering polytope

The linear ordering polytope P LO n is defined as the convex hull of the incidence vectors of the acyclic tournaments on n nodes. It is known that for every facet of P LO n , there corresponds a digraph inducing it. Let D be a digraph that induces a facet-defining inequality for P LO n , that is nonequivalent to a trivial inequality or to a 3-dicycle inequality. We show that for such a digraph the following holds: the value τ of a minimum integral dicycle cover is greater than the value τ ∗ of a minimum dicycle cover. We show that τ ∗ can be found by minimizing a linear function over a polytope which is defined by a polynomial number of constraints. Let v denote the value of a maximum integral dicycle packing. We prove that if D is a certain digraph with a two-node cut satisfying τ = v in each part, then τ = v in D as well. Dridi's description of P LO 5 enables a simple derivation of the fact that τ = v for any digraph on 5 nodes. Combining these results with the theorem of Lucchesi and Younger for planar digraphs as well as Wagner's decomposition, we obtain that τ = v in K 3.3-free digraphs. This last result was proved recently by Barahona et al. (1990) using polyhedral techniques while our proof is based mainly on combinatorial tools.

Surface Intersections for Geometric Modeling

Evaluation of planar algebraic curves arises in the context of intersections of algebraic surfaces with piecewise continuous rational polynomial parametric surface patches useful in geometric modeling. We address a method of evaluating these curves of intersection that combines the advantageous features of analytic representation of the governing equation of the algebraic curve in the Bernstein basis within a rectangular domain, adaptive subdivision and polyhedral faceting techniques, and the computation of turning and singular points, to provide the basis for a reliable and efficient solution procedure. Using turning and singular points, the intersection problem can be partitioned into subdomains that can be processed independently and which involve intersection segments that can be traced with faceting methods. This partitioning and the tracing of individual segments is carried out using an adaptive subdivision algorithm for Bezier/B-spline surfaces followed by Newton correction of the approximation. The method has been successfully tested in tracing complex algebraic curves and in solving actual intersection problems with diverse features.