Mirror-Descent Methods in Mixed-Integer Convex Optimization

In this paper, we address the problem of minimizing a convex function f over a convex set, with the extra constraint that some variables must be integer. This problem, even when f is a piecewise linear function, is NP-hard. We study an algorithmic approach to this problem, postponing its hardness to the realization of an oracle. If this oracle can be realized in polynomial time, then the problem can be solved in polynomial time as well. For problems with two integer variables, we show that the oracle can be implemented efficiently, that is, in O(ln(B)) approximate minimizations of f over the continuous variables, where B is a known bound on the absolute value of the integer variables.Our algorithm can be adapted to find the second best point of a purely integer convex optimization problem in two dimensions, and more generally its k-th best point. This observation allows us to formulate a finite-time algorithm for mixed-integer convex optimization

Socialist Dandies International: East Europe, 1946-1959

This article maps the looks and lifestyle choices of small groups of young, like-minded people who emerged in the postwar Soviet Union and East Europe in the background of huge political, social, and cultural changes. With their androgynous bodies wrapped in drape jackets and narrow trousers, and their love of jazz and swing, these young men stood in a sharp contrast to the official ideology that promoted socialism as a new, pure, and highly rationalized project, its ideal robust and strong man, and its mass culture that insisted on educational and restrained forms of entertainment. Through the categories of dress, body, and big city, the article investigates the clashes, and the eventual truce, between the socialist streamlined and rationalized master narrative and the young dandies' fragmented and disordered narrative. The article argues that the socialist dandies were not politically minded, and that their challenge to the officially proclaimed values was informed by their adolescent recklessness and a general postwar desolation. They were declared state enemies because the socialist regimes did not allow for alternative types of modernity. Consequently, the authorities condemned the young dandies' looks and interests as cosmopolitan, because they originated in the West, and as artificial, since they belonged to the culture that had preceded a new socialist world

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Local Hardy Spaces of Musielak-Orlicz Type and Their Applications

Let \phi: \mathbb{R}^n\times[0,\fz)\rightarrow[0,\fz) be a function such that $\phi(x,\cdot)$ is an Orlicz function and $\phi(\cdot,t)\in A^{\mathop\mathrm{loc}}_{\infty}(\mathbb{R}^n)$ (the class of local weights introduced by V. S. Rychkov). In this paper, the authors introduce a local Hardy space $h_{\phi}(\mathbb{R}^n)$ of Musielak-Orlicz type by the local grand maximal function, and a local $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}_{\phi}(\mathbb{R}^n)$ which is further proved to be the dual space of $h_{\phi}(\mathbb{R}^n)$. As an application, the authors prove that the class of pointwise multipliers for the local $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}^{\phi}(\mathbb{R}^n)$, characterized by E. Nakai and K. Yabuta, is just the dual of L^1(\rn)+h_{\Phi_0}(\mathbb{R}^n), where $\phi$ is an increasing function on $(0,\infty)$ satisfying some additional growth conditions and $\Phi_0$ a Musielak-Orlicz function induced by $\phi$. Characterizations of $h_{\phi}(\mathbb{R}^n)$, including the atoms, the local vertical and the local nontangential maximal functions, are presented. Using the atomic characterization, the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of $h_{\phi}(\mathbb{R}^n)$, from which, the authors further deduce some criterions for the boundedness on $h_{\phi}(\mathbb{R}^n)$ of some sublinear operators. Finally, the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on $h_{\phi}(\mathbb{R}^n)$.Comment: Sci. China Math. (to appear

GraphCombEx: A Software Tool for Exploration of Combinatorial Optimisation Properties of Large Graphs

We present a prototype of a software tool for exploration of multiple combinatorial optimisation problems in large real-world and synthetic complex networks. Our tool, called GraphCombEx (an acronym of Graph Combinatorial Explorer), provides a unified framework for scalable computation and presentation of high-quality suboptimal solutions and bounds for a number of widely studied combinatorial optimisation problems. Efficient representation and applicability to large-scale graphs and complex networks are particularly considered in its design. The problems currently supported include maximum clique, graph colouring, maximum independent set, minimum vertex clique covering, minimum dominating set, as well as the longest simple cycle problem. Suboptimal solutions and intervals for optimal objective values are estimated using scalable heuristics. The tool is designed with extensibility in mind, with the view of further problems and both new fast and high-performance heuristics to be added in the future. GraphCombEx has already been successfully used as a support tool in a number of recent research studies using combinatorial optimisation to analyse complex networks, indicating its promise as a research software tool

Nonlinear spectral calculus and super-expanders

Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space.Comment: Typos fixed based on referee comments. Some of the results of this paper were announced in arXiv:0910.2041. The corresponding parts of arXiv:0910.2041 are subsumed by the current pape

Genetic Signature of Rapid IHHNV (Infectious Hypodermal and Hematopoietic Necrosis Virus) Expansion in Wild Penaeus Shrimp Populations

Infectious hypodermal and hematopoietic necrosis virus (IHHNV) is a widely distributed single-stranded DNA parvovirus that has been responsible for major losses in wild and farmed penaeid shrimp populations on the northwestern Pacific coast of Mexico since the early 1990's. IHHNV has been considered a slow-evolving, stable virus because shrimp populations in this region have recovered to pre-epizootic levels, and limited nucleotide variation has been found in a small number of IHHNV isolates studied from this region. To gain insight into IHHNV evolutionary and population dynamics, we analyzed IHHNV capsid protein gene sequences from 89 Penaeus shrimp, along with 14 previously published sequences. Using Bayesian coalescent approaches, we calculated a mean rate of nucleotide substitution for IHHNV that was unexpectedly high (1.39×10−4 substitutions/site/year) and comparable to that reported for RNA viruses. We found more genetic diversity than previously reported for IHHNV isolates and highly significant subdivision among the viral populations in Mexican waters. Past changes in effective number of infections that we infer from Bayesian skyline plots closely correspond to IHHNV epizootiological historical records. Given the high evolutionary rate and the observed regional isolation of IHHNV in shrimp populations in the Gulf of California, we suggest regular monitoring of wild and farmed shrimp and restriction of shrimp movement as preventative measures for future viral outbreaks