While semidefinite programming (SDP) problems are polynomially solvable in theory, it is often difficult to solve large SDP instances in practice. One technique to address this issue is to relax the global positive-semidefiniteness (PSD) constraint and only enforce PSD-ness on smaller $k\times k$ principal submatrices --- we call this the sparse SDP relaxation. Surprisingly, it has been observed empirically that in some cases this approach appears to produce bounds that are close to the optimal objective function value of the original SDP. In this paper, we formally attempt to compare the strength of the sparse SDP relaxation vis-à-vis the original SDP from a theoretical perspective.
In order to simplify the question, we arrive at a data independent version of it, where we compare the sizes of SDP cone and the $k$-PSD closure, which is the cone of matrices where PSD-ness is enforced on all $k\times k$ principal submatrices. In particular, we investigate the question of how far a matrix of unit Frobenius norm in the $k$-PSD closure can be from the SDP cone. We provide two incomparable upper bounds on this farthest distance as a function of $k$ and $n$. We also provide matching lower bounds, which show that the upper bounds are tight within a constant in different regimes of $k$ and $n$. Other than linear algebra techniques, we extensively use probabilistic methods to arrive at these bounds. One of the lower bounds is obtained by observing a connection between matrices in the $k$-PSD closure and matrices satisfying the restricted isometry property (RIP).
Approximate Projections onto the Positive Semidefinite Cone Using Randomization
School of Electrical and Electronic Engineering, The University of Sheffield（谢菲尔德大学电气与电子工程学院）；Department of Electrical Engineering, Columbia University（哥伦比亚大学电气工程系）
Exponential lower bounds on fixed-size psd rank and semidefinite extension complexity
最近有很多兴趣证明表示给定多面体P所需的线性规划的大小的下界。在Fiorini等人的突破性论文[Proceedings of 44 th ACM Symposium on Theory of Computing 2012，第95-106页]中，表明最大割的任何线性规划公式必须具有指数大小。一个自然的问题是，是否可以证明这样强的下界半定规划公式。在本文中，我们朝着这个目标迈出了一步，我们证明了强下界的某一类SDP配方，即SDPs的笛卡儿积的固定大小的半正定锥。在实践中，这对应于具有块对角结构的半定程序，其中块具有恒定大小d。我们表明，任何这样的扩展配方的切割多面体必须有指数的大小（当d是固定的）。Fiorini等人对LP公式的结果是d=1时的特殊情况。对于大小为d=2的块，结果排除了使用二阶锥规划的任何小公式。我们的研究SDP升降机上的笛卡儿积的固定大小的半正定锥的动机主要是从实际的考虑，众所周知，这样的SDP可以更有效地解决比一般的SDP。我们的下界的证明依赖于一些矩阵的稀疏模式与小psd秩的新结果，结合从最近的论文启发Kaibel和Weltge [arXiv：1307.3543]的相关多面体的LP扩展复杂性的归纳参数。
On Polyhedral and Second-Order Cone Decompositions of Semidefinite Optimization Problems
Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology（麻省理工学院斯隆管理学院与运筹学中心）；Operations Research Center, Massachusetts Institute of Technology（麻省理工学院运筹学中心）
Finding sparse solutions of systems of polynomial equations via group-sparsity optimization
LORIA（洛林信息学及其应用研究实验室）；Université de Lorraine（洛林大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；University of California, Berkeley（加州大学伯克利分校）；Linköping University（林雪平大学）
