The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Thank you for downloading linear and nonlinear programming. As you may know, people have search numerous times for their favorite novels like this linear and nonlinear programming, but end up in malicious downloads. Rather than reading a good book with a cup of tea in the afternoon, instead they juggled with some infectious bugs inside their desktop computer. linear and nonlinear programming is available in our book collection an online access to it is set as public so you can download it instantly. Our digital library spans in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the linear and nonlinear programming is universally compatible with any devices to read.

Linear And Nonlinear Programming

In this chapter we will introduce submodularity and some of its generalizations, illustrate how it arises in various applications, and discuss algorithms for optimizing submodular functions. Submodularity is a property of set functions with deep theoretical consequences and far-reaching applications. At first glance it seems very similar to concavity, in other ways it resembles convexity. It appears in a wide variety of applications: in Computer Science it has recently been identified and utilized in domains such as viral marketing [39], information gathering [44], image segmentation [10, 40, 36], document summarization [56], and speeding up satisfiability solvers [73]. Our emphasis in this chapter is on maximization; there are many important results and applications related to minimizing submodular functions that we do not cover. As a concrete running example, we will consider the problem of deploying sensors in a drinking water distribution network (see Figure 3.1) in order to detect contamination. In this domain, we may have a model of how contaminants, accidentally or maliciously introduced into the network, spread over time. Such a model then allows to quantify the benefit f(A) of deploying sensors at a particular set A of locations (junctions or pipes in the network) in terms of the detection performance (such as average time to detection). Based on this notion of utility, we then wish to find an optimal subset A ⊆ V of locations maximizing the utility, max A f(A) , subject to some constraints (such as bounded cost). This application requires solving a difficult real-world optimization problem, that can be handled with the techniques discussed in this chapter (Krause et al. [49] show in detail how submodular optimization can be applied in this domain.)

/pdf/submodular-function-maximization-1dos5dw1tz.pdf

Submodular Function Maximization

Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

Probability on Trees and Networks

Discrete optimization problems are everywhere, from traditional operations research planning problems, such as scheduling, facility location, and network design; to computer science problems in databases; to advertising issues in viral marketing. Yet most such problems are NP-hard. Thus unless P = NP, there are no efficient algorithms to find optimal solutions to such problems. This book shows how to design approximation algorithms: efficient algorithms that find provably near-optimal solutions. The book is organized around central algorithmic techniques for designing approximation algorithms, including greedy and local search algorithms, dynamic programming, linear and semidefinite programming, and randomization. Each chapter in the first part of the book is devoted to a single algorithmic technique, which is then applied to several different problems. The second part revisits the techniques but offers more sophisticated treatments of them. The book also covers methods for proving that optimization problems are hard to approximate. Designed as a textbook for graduate-level algorithms courses, the book will also serve as a reference for researchers interested in the heuristic solution of discrete optimization problems.

https://www.designofapproxalgs.com/book.pdf

The Design of Approximation Algorithms

In this paper we give the first approximation algorithm for the problem of max-min fair allocation of indivisible goods. The approximation ratio of our algorithm is Ω1√k log3 k. As a part of our algorithm, we design an iterative method for rounding a fractional matching on a tree which might be of independent interest.

https://web.stanford.edu/~saberi/maxmin.pdf

An approximation algorithm for max-min fair allocation of indivisible goods

We consider the Asymmetric Traveling Salesman problem for costs satisfying the triangle inequality. We derive a randomized algorithm which delivers a solution within a factor O(log n/ log log n) of the optimum with high probability.

/pdf/an-o-log-n-log-log-n-approximation-algorithm-for-the-u8rbnz0ccy.pdf

An O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem

An O(log n/log log n)-approximation algorithm for the asymmetric traveling salesman problem

In this paper, we give the first approximation algorithm for the problem of max-min fair allocation of indivisible goods. An instance of this problem consists of a set of $k$ people and $m$ indivisible goods. Each person has a known linear utility function over the set of goods which might be different from the utility functions of other people. The goal is to distribute the goods among the people and maximize the minimum utility received by them. The approximation ratio of our algorithm is $\Omega(\frac{1}{\sqrt{k}\log^{3}k})$. As a crucial part of our algorithm, we design and analyze an iterative method for rounding a fractional matching on a tree which might be of independent interest. We also provide better bounds when we are allowed to exclude a small fraction of the people from the problem.

An Approximation Algorithm for Max-Min Fair Allocation of Indivisible Goods

We present a randomized O(log n/log log n)-approximation algorithm for the asymmetric traveling salesman problem (ATSP) This provides the first asymptotic improvement over the long-standing Θ(log n)-approximation bound stemming from the work of Frieze et al (1982) [Frieze AM, Galbiati G, Maffioki F (1982) On the worst-case performance of some algorithms for the asymmetric traveling salesman problem Networks 12(1):23–39] The key ingredient of our approach is a new connection between the approximability of the ATSP and the notion of so-called thin trees To exploit this connection, we employ maximum entropy rounding—a novel method of randomized rounding of LP relaxations of optimization problems We believe that this method might be of independent interest

Arash Asadpour

Papers

An approximation algorithm for max-min fair allocation of indivisible goods

An O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem

An O(log n/log log n)-approximation algorithm for the asymmetric traveling salesman problem

An Approximation Algorithm for Max-Min Fair Allocation of Indivisible Goods

An O(log n/log log n)-Approximation Algorithm for the Asymmetric Traveling Salesman Problem