Based on the well-known result that the sum of the largest eigenvalues of a symmetric matrix can be represented as a semidefinite programming problem (SDP), we formulate the nearest low-rank correlation matrix problem as a nonconvex SDP and propose a numerical method that solves a sequence of least-square problems. Each of the least-square problems can be solved by a specifically designed semismooth Newton method, which is shown to be quadratically convergent. The sequential method is guaranteed to produce a stationary point of the nonconvex SDP. Our numerical results demonstrate the high efficiency of the proposed method on large scale problems.
Background Coronary heart disease has become a serious challenge to China with its high prevalence and mortality. The impaired glucose regulation is prevalent in patients with cardiovascular disease. However, there are few drugs that interfere early with impaired glucose regulation. Ginkgo biloba extract not only is a commonly used drug for cardiovascular diseases, but also has a significant effect in reducing blood sugar. Therefore, this study used a single-case randomized controlled trial to explore the efficacy of Ginkgo biloba pills in the treatment of coronary heart disease patients with impaired glucose regulation. Methods/Design This is a randomized, double-blind, placebo-controlled, three-period crossover trial for a single subject. A total of 12 subjects will be recruited in this trial. The trial is divided into three cycles, and one cycle has two treatment periods. Ginkgo biloba pills and placebo will be randomized during the treatment period. The test period will last for 58 weeks and subjects will take 48 weeks. Subjects will be selected by the researcher strictly in accordance with the inclusion and exclusion criteria. Discussion Ginkgo biloba preparations are widely used in cardiovascular diseases both at home and abroad due to their definite curative effect, few side effects, various dosage forms, and convenient and safe use. Diabetes mellitus is a high-risk factor for the occurrence of cardiovascular disease. Therefore, it is of great significance to control the impaired glucose regulation and slow down the development of diabetes and reduce the incidence and mortality of cardiovascular diseases. This trial is registered with ClinicalTrials.gov (ID: NCT03483779).
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