/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

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.

/pdf/scip-solving-constraint-integer-programs-21j9hcniww.pdf

SCIP: solving constraint integer programs

Gearboxes are mechanical transmission systems that provide speed and torque conversions from a rotating power source. Being a central element of the drive train, they are relevant for the efficiency and durability of motor vehicles. In this work, we present a new approach for gearbox design: Modeling the design problem as a mixed-integer nonlinear program (MINLP) allows us to create gearbox designs from scratch for arbitrary requirements and—given enough time—to compute provably globally optimal designs for a given objective. We show how different degrees of freedom influence the runtime and present an exemplary solution.

A Mixed-Integer Nonlinear Program for the Design of Gearboxes

Given the increasing availability of high-resolution input data, today's computer vision problems tend to grow beyond what has been considered tractable in the past. This is especially true for Markov Random Fields (MRFs), which have expanded beyond millions of variables with thousands of labels. Such MRFs pose new challenges for inference, requiring massively parallel solvers that can cope with large-scale problems and support general, irregular input graphs. We propose a block coordinate descent based solver for large MRFs designed to exploit many-core hardware such as recent GPUs. We identify tree-shaped subgraphs as a block coordinate scheme for irregular topologies and optimize them efficiently using dynamic programming. The resulting solver supports arbitrary MRF topologies efficiently and can handle arbitrary, dense or sparse label sets as well as label cost functions. Together with two additional heuristics for further acceleration, our solver performs favorably even compared to modern specialized solvers in terms of speed and solution quality, especially when solving very large MRFs.

A fast, massively parallel solver for large, irregular pairwise Markov random fields

Polynomial optimization problems over binary variables can be expressed as integer programs using a linearization with extra monomials in addition to those arising in the given polynomial. We characterize when such a linearization yields an integral relaxation polytope, generalizing work by Del Pia and Khajavirad (SIAM Journal on Optimization, 2018) and Buchheim, Crama and Rodriguez-Heck (European Journal of Operations Research, 2019). We also present an algorithm that finds these extra monomials for a given polynomial to yield an integral relaxation polytope or determines that no such set of extra monomials exists. In the former case, our approach yields an algorithm to solve the given polynomial optimization problem as a compact LP, and we complement this with a purely combinatorial algorithm.

Integrality of Linearizations of Polynomials over Binary Variables using Additional Monomials

We investigate the computational complexity of a problem arising in the context of sparse optimization, namely, the projection onto the set of k-cosparse vectors w.r.t. some given matrix Ω. We show that this projection problem is (strongly) NP-hard, even in the special cases where the matrix Ω contains only ternary or bipolar coefficients. Interestingly, this is in stark contrast to the projection onto the set of k-sparse vectors, which is trivially solved by keeping only the k largest coefficients.

https://hal.inria.fr/hal-00811671/document

Projection Onto The k-Cosparse Set is NP-Hard

/pdf/multicommodity-flows-and-column-generation-2acu9c06ts.pdf

Marc E. Pfetsch

Papers

A Mixed-Integer Nonlinear Program for the Design of Gearboxes

A fast, massively parallel solver for large, irregular pairwise Markov random fields

Integrality of Linearizations of Polynomials over Binary Variables using Additional Monomials

Projection Onto The k-Cosparse Set is NP-Hard

Multicommodity Flows and Column Generation