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Radon, the primary constituent of natural radiation, is the second leading environmental cause of lung cancer after smoking. To confirm a relationship between indoor radon exposure and lung cancer, estimating cumulative levels of exposure to indoor radon for an individual or population is necessary. This study sought to develop a model for estimate indoor radon concentrations in Korea. Especially, our model and method may have wider application to other residences, not to specific site, and can be used in situations where actual measurements for input variables are lacking. In order to develop a model, indoor radon concentrations were measured at 196 ground floor residences using passive alpha-track detectors between January and April 2016. The arithmetic mean (AM) and geometric mean (GM) means of indoor radon concentrations were 117.86±72.03 and 95.13±2.02 Bq/m3, respectively. Questionnaires were administered to assess the characteristics of each residence, the environment around the measuring equipment, and lifestyles of the residents. Also, national data on indoor radon concentrations at 7643 detached houses for 2011-2014 were reviewed to determine radon concentrations in the soil, and meteorological data on temperature and wind speed were utilized to approximate ventilation rates. The estimated ventilation rates and radon exhalation rates from the soil varied from 0.18 to 0.98/hr (AM, 0.59±0.17/hr) and 326.33 to 1392.77 Bq/m2/hr (AM, 777.45±257.39; GM, 735.67±1.40 Bq/m2/hr), respectively. With these results, the developed model was applied to estimate indoor radon concentrations for 157 residences (80% of all 196 residences), which were randomly sampled. The results were in better agreement for Gyeonggi and Seoul than for other regions of Korea. Overall, the actual and estimated radon concentrations were in better agreement, except for a few low-concentration residences.

This paper presents a duality theory for solving concave minimization problem and nonconvex quadratic programming problem subjected to nonlinear inequality constraints. By use of the canonical dual transformation developed recently, two canonical dual problems are formulated, respectively. These two dual problems are perfectly dual to the primal problems with zero duality gap. It is proved that the sufficient conditions for global minimizers and local extrema (both minima and maxima) are controlled by the triality theory discovered recently [5]. This triality theory can be used to develop certain useful primal-dual methods for solving difficult nonconvex minimization problems. Results shown that the difficult quadratic minimization problem with quadratic constraint can be converted into a one-dimensional dual problem, which can be solved completely to obtain all KKT points and global minimizer.