We extend the result on the spectral projected gradient method by Birgin et al. in 2000 to a log-determinant semidefinite problem with linear constraints and propose a spectral projected gradient method for the dual problem. Our method is based on alternate projections on the intersection of two convex sets, which first projects onto the box constraints and then onto a set defined by a linear matrix inequality. By exploiting structures of the two projections, we show that the same convergence properties can be obtained for the proposed method as Birgin’s method where the exact orthogonal projection onto the intersection of two convex sets is performed. Using the convergence properties, we prove that the proposed algorithm attains the optimal value or terminates in a finite number of iterations. The efficiency of the proposed method is illustrated with the numerical results on randomly generated synthetic/deterministic data and gene expression data, in comparison with other methods including the inexact primal–dual path-following interior-point method, the Newton-CG primal proximal-point algorithm, the adaptive spectral projected gradient method, and the adaptive Nesterov’s smooth method. For the gene expression data, our results are compared with the quadratic approximation for sparse inverse covariance estimation method. We show that our method outperforms the other methods in obtaining a better objective value fast.

中文翻译：

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对数行列式半定问题的双谱投影梯度法

我们将结果扩展到Birgin等人的光谱投影梯度法上。于2000年提出对数确定的具有线性约束的半确定问题，并提出了对偶问题的频谱投影梯度法。我们的方法基于两个凸集的交集上的交替投影，该投影首先投影到框约束上，然后投影到由线性矩阵不等式定义的集合上。通过利用两个投影的结构，我们表明，所提出的方法可以获得与Birgin方法相同的收敛性，在Birgin方法中，可以对两个凸集的交点进行精确的正交投影。利用收敛性，我们证明了所提出的算法达到了最佳值或终止了有限的迭代次数。与其他方法（包括不精确的原始-双路径跟踪内点方法，Newton-CG原始近端方法）相比，该方法的效率通过随机生成的合成/确定性数据和基因表达数据的数值结果得到说明。点算法，自适应频谱投影梯度方法和自适应Nesterov平滑方法。对于基因表达数据，将我们的结果与稀疏逆协方差估计方法的二次近似进行比较。我们表明，在快速获得更好的目标值方面，我们的方法优于其他方法。Newton-CG原始近端算法，自适应频谱投影梯度方法和自适应Nesterov平滑方法。对于基因表达数据，将我们的结果与稀疏逆协方差估计方法的二次近似进行比较。我们表明，在快速获得更好的目标值方面，我们的方法优于其他方法。Newton-CG原始近端算法，自适应频谱投影梯度方法和自适应Nesterov平滑方法。对于基因表达数据，将我们的结果与稀疏逆协方差估计方法的二次近似进行比较。我们表明，在快速获得更好的目标值方面，我们的方法优于其他方法。