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Background: Physical activity has a protective effect against cardiovascular disease (CVD) in high-income countries, where physical activity is mainly recreational, but it is not known if this is also observed in lower-income countries, where physical activity is mainly non-recreational. We examined whether different amounts and types of physical activity are associated with lower mortality and CVD in countries at different economic levels.Methods: In this prospective cohort study, we recruited participants from 17 countries (Canada, Sweden, United Arab Emirates, Argentina, Brazil, Chile, Poland, Turkey, Malaysia, South Africa, China, Colombia, Iran, Bangladesh, India, Pakistan, and Zimbabwe). Within each country, urban and rural areas in and around selected cities and towns were identified to reflect the geographical diversity. Within these communities, we invited individuals aged between 35 and 70 years who intended to live at their current address for at least another 4 years. Total physical activity was assessed using the International Physical Activity Questionnaire (IPQA). Participants with pre-existing CVD were excluded from the analyses. Mortality and CVD were recorded during a mean of 6·9 years of follow-up. Primary clinical outcomes during follow-up were mortality plus major CVD (CVD mortality, incident myocardial infarction, stroke, or heart failure), either as a composite or separately. The effects of physical activity on mortality and CVD were adjusted for sociodemographic factors and other risk factors taking into account household, community, and country clustering.Findings: Between Jan 1, 2003, and Dec 31, 2010, 168 916 participants were enrolled, of whom 141 945 completed the IPAQ. Analyses were limited to the 130 843 participants without pre-existing CVD. Compared with low physical activity (<600 metabolic equivalents [MET] × minutes per week or <150 minutes per week of moderate intensity physical activity), moderate (600-3000 MET × minutes or 150-750 minutes per week) and high physical activity (>3000 MET × minutes or >750 minutes per week) were associated with graded reduction in mortality (hazard ratio 0·80, 95% CI 0·74-0·87 and 0·65, 0·60-0·71; p<0·0001 for trend), and major CVD (0·86, 0·78-0·93; p<0·001 for trend). Higher physical activity was associated with lower risk of CVD and mortality in high-income, middle-income, and low-income countries. The adjusted population attributable fraction for not meeting the physical activity guidelines was 8·0% for mortality and 4·6% for major CVD, and for not meeting high physical activity was 13·0% for mortality and 9·5% for major CVD. Both recreational and non-recreational physical activity were associated with benefits.Interpretation: Higher recreational and non-recreational physical activity was associated with a lower risk of mortality and CVD events in individuals from low-income, middle-income, and high-income countries. Increasing physical activity is a simple, widely applicable, low cost global strategy that could reduce deaths and CVD in middle age.Funding: Population Health Research Institute, the Canadian Institutes of Health Research, Heart and Stroke Foundation of Ontario, Ontario SPOR Support Unit, Ontario Ministry of Health and Long-Term Care, AstraZeneca, Sanofi-Aventis, Boehringer Ingelheim, Servier, GSK, Novartis, King Pharma, and national and local organisations in participating countries that are listed at the end of the Article. [ABSTRACT FROM AUTHOR]

We consider a version of the knapsack problem in which an item size is random and revealed only when the decision maker attempts to insert it. After every successful insertion the decision maker can choose the next item dynamically based on the remaining capacity and available items, while an unsuccessful insertion terminates the process. We propose a new semi-infinite relaxation based on an affne value function approximation, and show that an existing pseudo-polynomial relaxation corresponds to a nonparametric value function approximation. We compare both theoretically to other relaxations from the literature and also perform a computational study. Our main empirical conclusion is that our new relaxation provides tight bounds over a variety of different instances and becomes tighter as the number of items increases. [ABSTRACT FROM AUTHOR]