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Security proof methods for quantum key distribution, QKD, that are based on the numerical key rate calculation problem, are powerful in principle. However, the practicality of the methods are limited by computational resources and the efficiency and accuracy of the underlying algorithms for convex optimization. We derive a stable reformulation of the convex nonlinear semidefinite programming, SDP, model for the key rate calculation problems. We use this to develop an efficient, accurate algorithm. The reformulation is based on novel forms of facial reduction, FR, for both the linear constraints and nonlinear relative entropy objective function. This allows for a Gauss-Newton type interior-point approach that avoids the need for perturbations to obtain strict feasibility, a technique currently used in the literature. The result is high accuracy solutions with theoretically proven lower bounds for the original QKD from the FR stable reformulation. This provides novel contributions for FR for general SDP.We report on empirical results that dramatically improve on speed and accuracy, as well as solving previously intractable problems.

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The purpose of this study is to assess the effects of heavy metal concentrations in the blood and urine of the general population. This research had been conducted from April to December 2008, studying 545 residents of Daejeon and Chungcheong Province. Through the concentrations of heavy metals(Pb, Cd, Hg, As, Mn) in the biota samples and questionnaires, the residents heavy metal exposure level and the influential factors according to personal characteristics or lifestyle were evaluated. As to the heavy metal concentration in the blood and urine of the comparing region, were As and Mn statistically significant(p<0.01, p<0.05). Blood lead and urinary mercury concentrations were higher in males than females. The heavy metal concentration for each age group increased blood mercury. The concentration of all heavy metals were higher in the drinkers than in the non-drinkers. Blood lead and mercury concentrations were higher in the smokers than in the non-smokers, but the urinary cadmium, arsenic and blood manganese was higher in the non-smokers than in the smokers. As to the blood lead and urinary cadmium concentration according to the food preference fish showed high concentration. To clarify the factors affecting the heavy metal concentration in biota among subjects multiple regression analysis was conducted. As a results, it turned out that as to lead content in blood, sex, age and smoking have influence on the subjects with explanatory adequacy of 14.0 %. These results demonstrated that the factors affected the concentrations of heavy metals in blood and urine. The results of this study could be used as the foundational data for setting the health risk assessment.

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Splitting methods in optimization arise when one can divide an optimization problem into two or more simpler subproblems. They have proven particularly successful for relaxations of problems involving discrete variables. We revisit and strengthen splitting methods for solving doubly nonnegative relaxations of the particularly difficult, NP-hard quadratic assignment problem. We use a modified restricted contractive splitting method approach. In particular, we show how to exploit redundant constraints in the subproblems. Our strengthened bounds exploit these new subproblems and new dual multiplier estimates to improve on the bounds and convergence results in the literature. Summary of Contribution: In our paper, we consider the quadratic assignment problem (QAP). It is one of the fundamental combinatorial optimization problems in the fields of optimization and operations research and includes many fundamental applications. We revisit and strengthen splitting methods for solving doubly nonnegative (DNN) relaxation of the QAP. We use a modified restricted contractive splitting method. We obtain strengthened bounds from improved lower and upper bounding techniques, and in fact, we solve many of these NP-hard problems to (provable) optimality, thus illustrating both the strength of the DNN relaxation and our new bounding techniques.

Security proof methods for quantum key distribution, QKD, that are based on the numerical key rate calculation problem, are powerful in principle. However, the practicality of the methods are limited by computational resources and the efficiency and accuracy of the underlying algorithms for convex optimization. We derive a stable reformulation of the convex nonlinear semidefinite programming, SDP, model for the key rate calculation problems. We use this to develop an efficient, accurate algorithm. The stable reformulation is based on novel forms of facial reduction, FR, for both the linear constraints and nonlinear quantum relative entropy objective function. This allows for a Gauss-Newton type interior-point approach that avoids the need for perturbations to obtain strict feasibility, a technique currently used in the literature. The result is high accuracy solutions with theoretically proven lower bounds for the original QKD from the FR stable reformulation. This provides novel contributions for FR for general SDP. We report on empirical results that dramatically improve on speed and accuracy, as well as solving previously intractable problems.

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