Simple Approximations of Semialgebraic Sets and their Applications to Control

Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance the solution set of linear matrix inequalities or the Schur/Hurwitz stability domains. These sets often have very complicated shapes (non-convex, and even non-connected), which renders very difficult their manipulation. It is therefore of considerable importance to find simple-enough approximations of these sets, able to capture their main characteristics while maintaining a low level of complexity. For these reasons, in the past years several convex approximations, based for instance on hyperrect-angles, polytopes, or ellipsoids have been proposed. In this work, we move a step further, and propose possibly non-convex approximations , based on a small volume polynomial superlevel set of a single positive polynomial of given degree. We show how these sets can be easily approximated by minimizing the L1 norm of the polynomial over the semialgebraic set, subject to positivity constraints. Intuitively, this corresponds to the trace minimization heuristic commonly encounter in minimum volume ellipsoid problems. From a computational viewpoint, we design a hierarchy of linear matrix inequality problems to generate these approximations, and we provide theoretically rigorous convergence results, in the sense that the hierarchy of outer approximations converges in volume (or, equivalently, almost everywhere and almost uniformly) to the original set. Two main applications of the proposed approach are considered. The first one aims at reconstruction/approximation of sets from a finite number of samples. In the second one, we show how the concept of polynomial superlevel set can be used to generate samples uniformly distributed on a given semialgebraic set. The efficiency of the proposed approach is demonstrated by different numerical examples

Randomized opinion dynamics over networks: influence estimation from partial observations

In this paper, we propose a technique for the estimation of the influence matrix in a sparse social network, in which $n$ individual communicate in a gossip way. At each step, a random subset of the social actors is active and interacts with randomly chosen neighbors. The opinions evolve according to a Friedkin and Johnsen mechanism, in which the individuals updates their belief to a convex combination of their current belief, the belief of the agents they interact with, and their initial belief, or prejudice. Leveraging recent results of estimation of vector autoregressive processes, we reconstruct the social network topology and the strength of the interconnections starting from \textit{partial observations} of the interactions, thus removing one of the main drawbacks of finite horizon techniques. The effectiveness of the proposed method is shown on randomly generation networks

Learning Hidden Influences in Large-Scale Dynamical Social Networks: A Data-Driven Sparsity-Based Approach, in Memory of Roberto Tempo

The processes of information diffusion across social networks (for example, the spread of opinions and the formation of beliefs) are attracting substantial interest in disciplines ranging from behavioral sciences to mathematics and engineering (see "Summary"). Since the opinions and behaviors of each individual are infl uenced by interactions with others, understanding the structure of interpersonal infl uences is a key ingredient to predict, analyze, and, possibly, control information and decisions [1]. With the rapid proliferation of social media platforms that provide instant messaging, blogging, and other networking services (see "Online Social Networks") people can easily share news, opinions, and preferences. Information can reach a broad audience much faster than before, and opinion mining and sentiment analysis are becoming key challenges in modern society [2]. The first anecdotal evidence of this fact is probably the use that the Obama campaign made of social networks during the 2008 U.S. presidential election [3]. More recently, several news outlets stated that Facebook users played a major role in spreading fake news that might have infl uenced the outcome of the 2016 U.S. presidential election [4]. This can be explained by the phenomena of homophily and biased assimilation [5]-[7] in social networks, which correspond to the tendency of people to follow the behaviors of their friends and establish relationships with like-minded individuals

A multicentre outcome analysis to define global benchmarks for donation after circulatory death liver transplantation

BACKGROUND: To identify the best possible outcomes in liver transplantation from donation after circulatory death donors (DCD) and to propose outcome values, which serve as reference for individual liver recipients or patient groups. METHODS: Based on 2219 controlled DCD liver transplantations, collected from 17 centres in North America and Europe, we identified 1012 low-risk, primary, adult liver transplantations with a laboratory MELD of ≤20points, receiving a DCD liver with a total donor warm ischemia time of ≤30minutes and asystolic donor warm ischemia time of ≤15minutes. Clinically relevant outcomes were selected and complications were reported according to the Clavien-Dindo-Grading and the Comprehensive Complication Index (CCI). Corresponding benchmark cut-offs were based on median values of each centre, where the 75(th)-percentile was considered. RESULTS: Benchmark cases represented between 19.7% and 75% of DCD transplantations in participating centers. The one-year retransplant and mortality rate was 5.23% and 9.01%, respectively. Within the first year of follow-up, 51.1% of recipients developed at least one major complication (≥Clavien-Dindo-Grade-III). Benchmark cut-offs were ≤3days and ≤16days for ICU and hospital stay, ≤66% for severe recipient complications (≥Grade-III), ≤16.8% for ischemic cholangiopathy, and ≤38.9CCI points at one-year posttransplant. Comparisons with higher risk groups showed more complications and impaired graft survival, outside the benchmark cut-offs. Organ perfusion techniques reduced the complications to values below benchmark cut-offs, despite higher graft risk. CONCLUSIONS: Despite excellent 1-year survival, morbidity in benchmark cases remains high with more than half of recipients developing severe complications during 1-year follow-up. Benchmark cut-offs targeting morbidity parameters offer a valid tool to assess the protective value of new preservation technologies in higher risk groups, and provide a valid comparator cohort for future clinical trials. LAY SUMMARY: The best possible outcomes after liver transplantation of grafts donated after circulatory death (DCD) were defined using the concept of benchmarking. These were based on 2219 liver transplantations following controlled DCD donation in 17 centres worldwide. The following benchmark cut-offs for the most relevant outcome parameters were developed: ICU and hospital stay: ≤3 and ≤16 days; primary non function: ≤2.5%; renal replacement therapy: ≤9.6%; ischemic cholangiopathy: ≤16.8% and anastomotic strictures ≤28.4%. One-year graft loss and mortality were defined as ≤14.4% and 9.6%, respectively. Donor and recipient combinations with higher risk had significantly worse outcomes. The use of novel organ perfusion technology achieved similar, good results in this high-risk group with prolonged donor warm ischemia time, when compared to the benchmark cohort