Gauge duality theory was originated by Preund （1987）, and was recently further investigated by Friedlander et al. （2014）. When solving some matrix optimization problems via gauge dual, one is usually able to avoid...Gauge duality theory was originated by Preund （1987）, and was recently further investigated by Friedlander et al. （2014）. When solving some matrix optimization problems via gauge dual, one is usually able to avoid full matrix decompositions such as singular value and/or eigenvalue decompositions. In such an approach, a gauge dual problem is solved in the first stage, and then an optimal solution to the primal problem can be recovered from the dual optimal solution obtained in the first stage. Recently, this theory has been applied to a class of semidefinite programming （SDP） problems with promising numerical results by Friedlander and Mac^to （2016）. We establish some theoretical results on applying the gauge duality theory to robust principal component analysis （PCA） and general SDP. For each problem, we present its gauge dual problem, characterize the optimality conditions for the primal-dual gauge pair, and validate a way to recover a primal optimal solution from a dual one. These results are extensions of Friedlander and Macedo （2016） from nuclear norm regularization to robust PCA and from a special class of SDP which requires the coefficient matrix in the linear objective to be positive definite to SDP problems without this restriction. Our results provide further understanding in the potential advantages and disadvantages of the gauge duality theory.展开更多
目的:对本实验室洛氏硬度计(C标尺)测量结果的不确定度作以评定。方法:按照JJG112-2003《金属洛氏硬度计(A B C D E F G H K N T标尺)检定规程》,以C标尺为例,考虑洛氏硬度试验中标准硬度块、硬度计及试样等因素对洛氏硬度计(C标尺)测...目的:对本实验室洛氏硬度计(C标尺)测量结果的不确定度作以评定。方法:按照JJG112-2003《金属洛氏硬度计(A B C D E F G H K N T标尺)检定规程》,以C标尺为例,考虑洛氏硬度试验中标准硬度块、硬度计及试样等因素对洛氏硬度计(C标尺)测量结果的不确定度进行评定,即通过分析、计算洛氏硬度试验中产生不确定度的若干分量,计算出扩展不确定度。结果:洛氏硬度计测量结果的扩展不确定度对32.5 HRC标准硬度块为:y_1=y_1±U_(95)=33.11HRC±1.078HRC,k=2;对62.3 HRC标准硬度块为:y_2=y_2±U_(95)=63.31HRC±0.710HRC,k=2。结论:在洛氏硬度计检定过程中,测量结果的不确定度主要取决于洛氏硬度计的允许误差。评定结果令人满意,符合洛氏硬度(C标尺)的测量要求。展开更多
基金supported by Hong Kong Research Grants Council General Research Fund (Grant No. 14205314)National Natural Science Foundation of China (Grant No. 11371192)
文摘Gauge duality theory was originated by Preund （1987）, and was recently further investigated by Friedlander et al. （2014）. When solving some matrix optimization problems via gauge dual, one is usually able to avoid full matrix decompositions such as singular value and/or eigenvalue decompositions. In such an approach, a gauge dual problem is solved in the first stage, and then an optimal solution to the primal problem can be recovered from the dual optimal solution obtained in the first stage. Recently, this theory has been applied to a class of semidefinite programming （SDP） problems with promising numerical results by Friedlander and Mac^to （2016）. We establish some theoretical results on applying the gauge duality theory to robust principal component analysis （PCA） and general SDP. For each problem, we present its gauge dual problem, characterize the optimality conditions for the primal-dual gauge pair, and validate a way to recover a primal optimal solution from a dual one. These results are extensions of Friedlander and Macedo （2016） from nuclear norm regularization to robust PCA and from a special class of SDP which requires the coefficient matrix in the linear objective to be positive definite to SDP problems without this restriction. Our results provide further understanding in the potential advantages and disadvantages of the gauge duality theory.
文摘目的:对本实验室洛氏硬度计(C标尺)测量结果的不确定度作以评定。方法:按照JJG112-2003《金属洛氏硬度计(A B C D E F G H K N T标尺)检定规程》,以C标尺为例,考虑洛氏硬度试验中标准硬度块、硬度计及试样等因素对洛氏硬度计(C标尺)测量结果的不确定度进行评定,即通过分析、计算洛氏硬度试验中产生不确定度的若干分量,计算出扩展不确定度。结果:洛氏硬度计测量结果的扩展不确定度对32.5 HRC标准硬度块为:y_1=y_1±U_(95)=33.11HRC±1.078HRC,k=2;对62.3 HRC标准硬度块为:y_2=y_2±U_(95)=63.31HRC±0.710HRC,k=2。结论:在洛氏硬度计检定过程中,测量结果的不确定度主要取决于洛氏硬度计的允许误差。评定结果令人满意,符合洛氏硬度(C标尺)的测量要求。
