We extend the result on the spectral projected gradient method by Birgin et al. in 2000 to a log-determinant semidefinite problem (SDP) 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 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 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 optimal value fast.
An Inexact Projected Gradient Method with Rounding and Lifting by Nonlinear Programming for Solving Rank-One Semidefinite Relaxation of Polynomial Optimization
Massachusetts Institute of Technology（麻省理工学院）；National University of Singapore（新加坡国立大学）
Eigenvalue, Quadratic Programming, and Semidefinite Programming Bounds for a Cut Minimization Problem
考虑将图的节点集划分为给定大小的$k$集，从而得到使用（去除）第$k$集得到的\emph{minimize the cut}的问题.如果得到的割具有值$0$，那么我们得到了顶点分隔符。这个问题与图的划分问题密切相关。事实上，我们使用的模型与图划分问题的模型相同，除了不同的\emph{quadratic}目标函数。我们期待在已知的和新的界限，从各种放宽这个NP-困难的问题。这包括：标准特征值界、使用邻接矩阵和拉普拉斯算子的投影特征值界、基于最近成功的二次指派问题的二次规划（QP）界的QP界、以及半定规划界。我们包括大型和\emph{huge}的问题，说明在强度和时间方面的边界的效率的数值测试。
Convex Relaxation for Solving Large-Margin Classifiers in Hyperbolic Space
John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保尔森工程与应用科学学院）；Center for Brain Science, Harvard University（哈佛大学脑科学中心）；Kempner Institute for the Study of Natural and Artificial Intelligence, Harvard University（哈佛大学肯普ner自然与人工智能研究所）
