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Positive Semidefinite Matrix and Cone Sn={A∈Rn×n:A⊤=A,A⪰0} 锥(cone) Definition A set C is a cone if x\in C implies \alpha x\in C for any \alpha\geq 0 Proof \alpha\geq 0,x^\top Ax\geq0\Rightarrow x^\top\alpha Ax=\alpha x^\top Ax\geq0 闭(closed) Definition ...
Hence the positive semidefinite cone is convex. It is a unique immutable proper cone in the ambient space of symmetric matrices. The positive definite(full-rank)matrices comprise the cone interior, while all singular positive semidefinite matrices(having at least oneeigenvalue)reside on the cone bou...
We study the tropicalization of the cone of positive semidefinite matrices over the ordered field of real Puiseux series. The tropical PSD matrices form the normal cone of the Newton polytope of the symmetric determinant at the vertex corresponding to the product of diagonal entries. We find ...
T-semidefinite coneT-semidefinite programmingPolynomial optimizationThe T-product for third-order tensors has been used extensively in the literature. In this paper, we first introduce first-order and second-order T-derivatives for the multi-variable real-valued function with the tensor T-product. ...
Positive semidefinite coneThresholding operatorGlobal solutionFixed pointThe [equation]–[equation] minimization over the positive semidefinite cone is the semidefinite least squares problem with Schatten [equation]-quasi ([equation]) norm regularization term. It has wide......
The cone CP n,q of completely positive linear transformations from M n ( C )= M n to M q is shown to be isometrically isomorphic to P nq , the cone of nq by nq positive semidefinite matrices. Generalizations of scalar and matrix results to CP n, q HP n, q L ( M n , M q ...
It is shown that an upper bound for the rank of the solution to a system of HLME over the positive semidefinite cone can be obtained efficiently by solving a semidefinite programming (SDP) problem. Moreover, a sufficient condition for the nonexistence of a rank-one solution to the system ...
Jingyong TangDepartment of MathematicsJein-Shan ChenDepartment of MathematicsOptimizationJ.C. Zhou, J.Y. Tang and J.-S. Chen (2016). Further relationship between second-order cone and positive semidefinite cone. Optimization 65(12):2115-2133....
This paper concerns the cone of positive semidefinite matrices which have zeros in prescribed entries. One of the main purposes is to obtain information and, if possible, describe the ranks of external matrices in such cones in terms of the pattern of prescribed zeros. In particular, we study ...