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Convex Analysisの二,三の進展について

From the Publisher:
Discrete geometry investigates combinatorial properties of configurations of geometric objects. To a working mathematician or computer scientist, it offers sophisticated results and techniques of great diversity and it is a foundation for fields such as computational geometry or combinatorial optimization.
This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and
high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces.
Jiri Matousek is Professor of Computer Science at Charles University in Prague. His research has contributed to several of the considered areas and to their algorithmic applications. This is his third book.

Lectures on Discrete Geometry

Computational geometry

The vast majority of today's critical infrastructure is supported by numerous feedback control loops and an attack on these control loops can have disastrous consequences. This is a major concern since modern control systems are becoming large and decentralized and thus more vulnerable to attacks. This paper is concerned with the estimation and control of linear systems when some of the sensors or actuators are corrupted by an attacker. We give a new simple characterization of the maximum number of attacks that can be detected and corrected as a function of the pair $(A,C)$ of the system and we show in particular that it is impossible to accurately reconstruct the state of a system if more than half the sensors are attacked. In addition, we show how the design of a secure local control loop can improve the resilience of the system. When the number of attacks is smaller than a threshold, we propose an efficient algorithm inspired from techniques in compressed sensing to estimate the state of the plant despite attacks. We give a theoretical characterization of the performance of this algorithm and we show on numerical simulations that the method is promising and allows to reconstruct the state accurately despite attacks. Finally, we consider the problem of designing output-feedback controllers that stabilize the system despite sensor attacks. We show that a principle of separation between estimation and control holds and that the design of resilient output feedback controllers can be reduced to the design of resilient state estimators.

Secure Estimation and Control for Cyber-Physical Systems Under Adversarial Attacks

Constraint integer programming (CIP) is a novel paradigm which integrates constraint programming (CP), mixed integer programming (MIP), and satisfiability (SAT) modeling and solving techniques. In this paper we discuss the software framework and solver SCIP (Solving Constraint Integer Programs), which is free for academic and non-commercial use and can be downloaded in source code. This paper gives an overview of the main design concepts of SCIP and how it can be used to solve constraint integer programs. To illustrate the performance and flexibility of SCIP, we apply it to two different problem classes. First, we consider mixed integer programming and show by computational experiments that SCIP is almost competitive to specialized commercial MIP solvers, even though SCIP supports the more general constraint integer programming paradigm. We develop new ingredients that improve current MIP solving technology. As a second application, we employ SCIP to solve chip design verification problems as they arise in the logic design of integrated circuits. This application goes far beyond traditional MIP solving, as it includes several highly non-linear constraints, which can be handled nicely within the constraint integer programming framework. We show anecdotally how the different solving techniques from MIP, CP, and SAT work together inside SCIP to deal with such constraint classes. Finally, experimental results show that our approach outperforms current state-of-the-art techniques for proving the validity of properties on circuits containing arithmetic.

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SCIP: solving constraint integer programs

Pseudo-Boolean problems generalize SAT problems by allowing linear constraints and a linear objective function. Different solvers, mainly having their roots in the SAT domain, have been proposed and compared,for instance, in Pseudo-Boolean evaluations. One can also formulate Pseudo-Boolean models as integer programming models. That is,Pseudo-Boolean problems lie on the border between the SAT domain and the integer programming field. In this paper, we approach Pseudo-Boolean problems from the integer programming side. We introduce the framework SCIP that implements constraint integer programming techniques. It integrates methods from constraint programming, integer programming, and SAT-solving: the solution of linear programming relaxations, propagation of linear as well as nonlinear constraints, and conflict analysis. We argue that this approach is suitable for Pseudo-Boolean instances containing general linear constraints, while it is less efficient for pure SAT problems. We present extensive computational experiments on the test set used for the Pseudo-Boolean evaluation 2007. We show that our approach is very efficient for optimization instances and competitive for feasibility problems. For the nonlinear parts, we also investigate the influence of linear programming relaxations and propagation methods on the performance. It turns out that both techniques are helpful for obtaining an efficient solution method.

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Solving Pseudo-Boolean Problems with SCIP

SOS1 constraints require that at most one of a given set of variables is nonzero. In this article, we investigate a branch-and-cut algorithm to solve linear programs with SOS1 constraints. We focus on the case in which the SOS1 constraints overlap. The corresponding conflict graph can algorithmically be exploited, for instance, for improved branching rules, preprocessing, primal heuristics, and cutting planes. In an extensive computational study, we evaluate the components of our implementation on instances for three different applications. We also demonstrate the effectiveness of this approach by comparing it to the solution of a mixed-integer programming formulation, if the variables appearing in SOS1 constraints ar bounded.

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Branch-and-cut for linear programs with overlapping SOS1 constraints

We investigate conditions for the unique recoverability of sparse integer-valued signals from a small number of linear measurements. Both the objective of minimizing the number of nonzero components, the so-called $\ell_0$-norm, as well as its popular substitute, the $\ell_1$-norm, are covered. Furthermore, integrality constraints and possible bounds on the variables are investigated. Our results show that the additional prior knowledge of signal integrality allows for recovering more signals than what can be guaranteed by the established recovery conditions from (continuous) compressed sensing. Moreover, even though the considered problems are \NP-hard in general (even with an $\ell_1$-objective), we investigate testing the $\ell_0$-recovery conditions via some numerical experiments. It turns out that the corresponding problems are quite hard to solve in practice using black-box software. However, medium-sized instances of $\ell_0$- and $\ell_1$-minimization with binary variables can be solved exactly within reasonable time.

Sparse Recovery With Integrality Constraints

The fare planning problem for public transport is to design a system of fares that maximize the revenue. We introduce a nonlinear optimization model to approach this problem. It is based on a discrete choice logit model that expresses demand as a function of the fares. We illustrate our approach by computing and comparing two different fare systems for the intercity network of the Netherlands.

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Optimal Fares for Public Transport

How much of the combinatorial structure of a pointed polyhedron is contained in its vertex-facet incidences? Not too much, in general, as we demonstrate by examples. However, one can tell from the incidence data whether the polyhedron is bounded. In the case of a poly- hedron that is simple and ''simplicial,'' i.e., a d-dimensional polyhedron that has d facets through each vertex and d vertices on each facet, we derive from the structure of the vertex- facet incidence matrix that the polyhedron is necessarily bounded. In particular, this yields a characterization of those polyhedra that have circulants as vertex-facet incidence matrices.

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Marc E. Pfetsch

Papers

Solving Pseudo-Boolean Problems with SCIP

Branch-and-cut for linear programs with overlapping SOS1 constraints

Sparse Recovery With Integrality Constraints

Optimal Fares for Public Transport

Vertex-Facet Incidences of Unbounded Polyhedra