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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

The development of mathematical simulation and optimization models and algorithms for solving gas transport problems is an active field of research. In order to test and compare these models and algorithms, gas network instances together with demand data are needed. The goal of GasLib is to provide a set of publicly available gas network instances that can be used by researchers in the field of gas transport. The advantages are that researchers save time by using these instances and that different models and algorithms can be compared on the same specified test sets. The library instances are encoded in an XML format. In this paper, we explain this format and present the instances that are available in the library.

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GasLib - A Library of Gas Network Instances

Chapter 12: Computational results for validation of nominations.

A new concept is introduced for the adaptive finite element discretization of partial differential equations that have a sparsely representable solution. Motivated by recent work on compressed sensing, a recursive mesh refinement procedure is presented that uses linear programming to find a good approximation to the sparse solution on a given refinement level. Then only those parts of the mesh are refined that belong to large expansion coefficients. Error estimates for this procedure are refined and the behavior of the procedure is demonstrated via some simple elliptic model problems.

Sparse Approximate Solution of Partial Differential Equations

The goal of this chapter is to define different types of uncertainty in technical systems and to provide a unified terminology for this book. Indeed, uncertainty comes in different disguises. The first distinction is made with respect to the knowledge on the source of uncertainty: stochastic uncertainty, incertitude or ignorance. Then three main occurrences of uncertainty are discussed: data, model and structural uncertainty.

Marc E. Pfetsch

Papers

GasLib - A Library of Gas Network Instances

Chapter 12: Computational results for validation of nominations.

Sparse Approximate Solution of Partial Differential Equations

Types of Uncertainty

Heuristic and Exact Methods for the Open Stacks Problems