Mathematical Programming, ISSN 0025-5610, 9/2018, Volume 171, Issue 1, pp. 397 - 431

An $$n\times n$$ n×n matrix X is called completely positive semidefinite (cpsd) if there exist $$d\times d$$ d×d Hermitian positive semidefinite matrices...

cpsd-rank | Quantum behaviors | Elliptope | 05C50 | Bell scenario | Theoretical, Mathematical and Computational Physics | 15B48 | Mathematics | Lorentz cone | Mathematical Methods in Physics | 81P45 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Quantum correlations | 90C25 | Numerical Analysis | 15A66 | Combinatorics | Completely positive semidefinite cone | 81P40 | MATHEMATICS, APPLIED | APPROXIMATIONS | CONIC FORMULATIONS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | PROGRAMS | MATRICES | OPTIMIZATION | Lower bounds | Matrix algebra | Matrix representation | Mathematical analysis | Upper bounds | Graphs | Matrix methods | Quantum theory

cpsd-rank | Quantum behaviors | Elliptope | 05C50 | Bell scenario | Theoretical, Mathematical and Computational Physics | 15B48 | Mathematics | Lorentz cone | Mathematical Methods in Physics | 81P45 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Quantum correlations | 90C25 | Numerical Analysis | 15A66 | Combinatorics | Completely positive semidefinite cone | 81P40 | MATHEMATICS, APPLIED | APPROXIMATIONS | CONIC FORMULATIONS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | PROGRAMS | MATRICES | OPTIMIZATION | Lower bounds | Matrix algebra | Matrix representation | Mathematical analysis | Upper bounds | Graphs | Matrix methods | Quantum theory

Journal Article

Mathematical Programming, ISSN 0025-5610, 03/2010, Volume 122, Issue 1, pp. 21 - 64

Let S = {x is an element of R-n : g(1)(x) >= 0, ..., g(m)(x) >= 0} be a semialgebraic set defined by multivariate polynomials g(i) (x). Assume S is convex,...

Convex sets | Modified Hessian | Sum of squares(SOS) | Semialgebraic geometry | Moments | Schmüdgen and Putinar's matrix Positivstellensatz | Convex polynomials positive curvature | Linear matrix inequality (LMI) | Semidefinite programming (SDP) | Poscurv-convex | MATHEMATICS, APPLIED | Schmudgen and Putinar's matrix Positivstellensatz | Extendable poscurv-convex | POLYNOMIALS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | Sos-concave (sos-convex) | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | POSITIVSTELLENSATZ | COMPLEXITY | Positive second fundamental form | Positive definite Lagrange Hessian (PDLH) condition | SQUARES | Mathematical optimization

Convex sets | Modified Hessian | Sum of squares(SOS) | Semialgebraic geometry | Moments | Schmüdgen and Putinar's matrix Positivstellensatz | Convex polynomials positive curvature | Linear matrix inequality (LMI) | Semidefinite programming (SDP) | Poscurv-convex | MATHEMATICS, APPLIED | Schmudgen and Putinar's matrix Positivstellensatz | Extendable poscurv-convex | POLYNOMIALS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | Sos-concave (sos-convex) | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | POSITIVSTELLENSATZ | COMPLEXITY | Positive second fundamental form | Positive definite Lagrange Hessian (PDLH) condition | SQUARES | Mathematical optimization

Journal Article

Discrete Applied Mathematics, ISSN 0166-218X, 01/2016, Volume 198, pp. 274 - 290

Let G be a simple, undirected graph. Positive semidefinite (PSD) zero forcing on G is based on the following color-change rule: Let W1,W2,…,Wk be the sets of...

Zero forcing number | Positive semidefinite | Minimum rank | Matrix | Graph | Maximum nullity | ZERO | MATHEMATICS, APPLIED

Zero forcing number | Positive semidefinite | Minimum rank | Matrix | Graph | Maximum nullity | ZERO | MATHEMATICS, APPLIED

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 01/2017, Volume 513, pp. 122 - 148

A real symmetric matrix M is completely positive semidefinite if it admits a Gram representation by (Hermitian) positive semidefinite matrices of any size d....

Hadamard matrices | Matrix factorization | Clifford algebras | Quantum correlations | Completely positive semidefinite cone | GRAPH | MATHEMATICS | MATHEMATICS, APPLIED | OPTIMIZATION | CONE | Mathematics - Optimization and Control

Hadamard matrices | Matrix factorization | Clifford algebras | Quantum correlations | Completely positive semidefinite cone | GRAPH | MATHEMATICS | MATHEMATICS, APPLIED | OPTIMIZATION | CONE | Mathematics - Optimization and Control

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 10/2013, Volume 439, Issue 7, pp. 1862 - 1874

The positive semidefinite zero forcing number Z+(G) of a graph G was introduced in Barioli et al. (2010) [4]. We establish a variety of properties of Z+(G):...

Zero forcing number | Positive semidefinite | Minimum rank | Matrix | Graph | Maximum nullity | MINIMUM-RANK | MATHEMATICS, APPLIED | NUMBER | NULLITY | GRAPHS

Zero forcing number | Positive semidefinite | Minimum rank | Matrix | Graph | Maximum nullity | MINIMUM-RANK | MATHEMATICS, APPLIED | NUMBER | NULLITY | GRAPHS

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 07/2019, Volume 572, pp. 51 - 67

It is obtained that for positive semidefinite block matrix H=[MKK⁎N] if A,B∈Mn(C) such that max(‖A‖2,‖B‖2)≤2 and f is a nonnegative increasing convex function...

Eigenvalue | Unitarily invariant norm | Convex function | Positive semidefinite matrix | Majorization | Inequality | Eigenvalues | Mathematical analysis | Matrix methods

Eigenvalue | Unitarily invariant norm | Convex function | Positive semidefinite matrix | Majorization | Inequality | Eigenvalues | Mathematical analysis | Matrix methods

Journal Article

SIAM Journal on Optimization, ISSN 1052-6234, 2010, Volume 20, Issue 5, pp. 2327 - 2351

We propose an algorithm for solving optimization problems defined on a subset of the cone of symmetric positive semidefinite matrices. This algorithm relies on...

Large-scale algorithms | Cone of symmetric positive definite matrices | Riemannian quotient manifold | Low-rank constraints | Maximum-cut algorithms | Sparse principal component analysis | low-rank constraints | large-scale algorithms | MATHEMATICS, APPLIED | PROGRAMS | cone of symmetric positive definite matrices | COMPONENT ANALYSIS | maximum-cut algorithms | sparse principal component analysis | SPECTRAL FUNCTIONS | Studies | Geometry | Optimization algorithms | Linear programming

Large-scale algorithms | Cone of symmetric positive definite matrices | Riemannian quotient manifold | Low-rank constraints | Maximum-cut algorithms | Sparse principal component analysis | low-rank constraints | large-scale algorithms | MATHEMATICS, APPLIED | PROGRAMS | cone of symmetric positive definite matrices | COMPONENT ANALYSIS | maximum-cut algorithms | sparse principal component analysis | SPECTRAL FUNCTIONS | Studies | Geometry | Optimization algorithms | Linear programming

Journal Article

Computational Optimization and Applications, ISSN 0926-6003, 9/2018, Volume 71, Issue 1, pp. 193 - 219

This paper considers the problem of positive semidefinite factorization (PSD factorization), a generalization of exact nonnegative matrix factorization. Given...

Positive semidefinite factorization | Extended formulations | Operations Research/Decision Theory | Convex and Discrete Geometry | Coordinate descent method | Mathematics | Operations Research, Management Science | Statistics, general | Fast gradient method | Optimization | MATHEMATICS, APPLIED | HIERARCHICAL ALS ALGORITHMS | RANK | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | PROGRAMS | COORDINATE DESCENT | COMPLEXITY | OPTIMIZATION | NONNEGATIVE MATRIX FACTORIZATION | Management science | Algorithms | Polyhedrons | Mathematical analysis | Matrix methods | Factorization | Local optimization

Positive semidefinite factorization | Extended formulations | Operations Research/Decision Theory | Convex and Discrete Geometry | Coordinate descent method | Mathematics | Operations Research, Management Science | Statistics, general | Fast gradient method | Optimization | MATHEMATICS, APPLIED | HIERARCHICAL ALS ALGORITHMS | RANK | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | PROGRAMS | COORDINATE DESCENT | COMPLEXITY | OPTIMIZATION | NONNEGATIVE MATRIX FACTORIZATION | Management science | Algorithms | Polyhedrons | Mathematical analysis | Matrix methods | Factorization | Local optimization

Journal Article

SIAM Journal on Optimization, ISSN 1052-6234, 2017, Volume 27, Issue 2, pp. 986 - 1009

The singularity degree of a semidefinite programming problem is the smallest number of facial reduction steps to make the problem strictly feasible. We...

Positive semidefinite matrix completion problem | Singularity degree | Universal rigidity | Uniquely solvable SDP | Facial reduction | Graph rigidity | graph rigidity | MATHEMATICS, APPLIED | facial reduction | singularity degree | uniquely solvable SDP | positive semidefinite matrix completion problem | universal rigidity | GRAPHS | FRAMEWORKS

Positive semidefinite matrix completion problem | Singularity degree | Universal rigidity | Uniquely solvable SDP | Facial reduction | Graph rigidity | graph rigidity | MATHEMATICS, APPLIED | facial reduction | singularity degree | uniquely solvable SDP | positive semidefinite matrix completion problem | universal rigidity | GRAPHS | FRAMEWORKS

Journal Article

Positivity, ISSN 1385-1292, 11/2018, Volume 22, Issue 5, pp. 1311 - 1324

In this paper, we propose three new matrix versions of the arithmetic–geometric mean inequality for unitarily invariant norms, which stem from the fact that...

Hilbert–Schmidt norm | Singular value | Mathematics | 47A30 | Secondary 15A18 | Trace | 15A42 | Operator Theory | Unitarily invariant norm | Fourier Analysis | Potential Theory | Calculus of Variations and Optimal Control; Optimization | Primary 15A60 | 47B15 | Positive semidefinite matrix | Econometrics | Inequality | MATHEMATICS | ANDO | HIAI | TRACE INEQUALITY | OKUBO | Hilbert-Schmidt norm | Norms | Real numbers | Mathematical analysis | Matrix methods | Inequalities | Arithmetic

Hilbert–Schmidt norm | Singular value | Mathematics | 47A30 | Secondary 15A18 | Trace | 15A42 | Operator Theory | Unitarily invariant norm | Fourier Analysis | Potential Theory | Calculus of Variations and Optimal Control; Optimization | Primary 15A60 | 47B15 | Positive semidefinite matrix | Econometrics | Inequality | MATHEMATICS | ANDO | HIAI | TRACE INEQUALITY | OKUBO | Hilbert-Schmidt norm | Norms | Real numbers | Mathematical analysis | Matrix methods | Inequalities | Arithmetic

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 05/2017, Volume 520, pp. 32 - 43

We bring in some new notions associated with 2×2 block positive semidefinite matrices. These notions concern the inequalities between the singular values of...

Eigenvalue | Positive partial transpose | Singular value | Inequality | MATHEMATICS | MATHEMATICS, APPLIED | INEQUALITIES | BLOCK | MAP

Eigenvalue | Positive partial transpose | Singular value | Inequality | MATHEMATICS | MATHEMATICS, APPLIED | INEQUALITIES | BLOCK | MAP

Journal Article

SIAM JOURNAL ON OPTIMIZATION, ISSN 1052-6234, 2015, Volume 25, Issue 2, pp. 1160 - 1178

We consider the projected semidefinite and Euclidean distance cones onto a subset of the matrix entries. These two sets are precisely the input data defining...

closedness | MATHEMATICS, APPLIED | matrix completion | CONVEX-SETS | Euclidean distance matrices | FACES | semidefinite programming | facial reduction | IMAGE | COMPLETION PROBLEMS | projection | POSITIVE SEMIDEFINITE | CONE | Slater condition

closedness | MATHEMATICS, APPLIED | matrix completion | CONVEX-SETS | Euclidean distance matrices | FACES | semidefinite programming | facial reduction | IMAGE | COMPLETION PROBLEMS | projection | POSITIVE SEMIDEFINITE | CONE | Slater condition

Journal Article

Mathematical Programming, ISSN 0025-5610, 9/2011, Volume 129, Issue 1, pp. 33 - 68

A basic framework for exploiting sparsity via positive semidefinite matrix completion is presented for an optimization problem with linear and nonlinear matrix...

Polynomial Optimization | Theoretical, Mathematical and Computational Physics | Matrix Inequalities | Mathematics | 90C26 | Chordal Graph | Mathematical Methods in Physics | 90C30 | Sparsity | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | 90C22 | Semidefinite Program | Combinatorics | Positive Semidefinite Matrix Completion | MATHEMATICS, APPLIED | SUMS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SDP-RELAXATIONS | SQUARES | POLYNOMIAL OPTIMIZATION PROBLEMS | Mathematical optimization | Studies | Optimization | Mathematical programming | Correlation | Mathematical analysis | Classification | Inequalities | Nonlinearity | Graphs | Conversion

Polynomial Optimization | Theoretical, Mathematical and Computational Physics | Matrix Inequalities | Mathematics | 90C26 | Chordal Graph | Mathematical Methods in Physics | 90C30 | Sparsity | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | 90C22 | Semidefinite Program | Combinatorics | Positive Semidefinite Matrix Completion | MATHEMATICS, APPLIED | SUMS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SDP-RELAXATIONS | SQUARES | POLYNOMIAL OPTIMIZATION PROBLEMS | Mathematical optimization | Studies | Optimization | Mathematical programming | Correlation | Mathematical analysis | Classification | Inequalities | Nonlinearity | Graphs | Conversion

Journal Article

ACM Transactions on Algorithms (TALG), ISSN 1549-6325, 02/2016, Volume 12, Issue 1, pp. 1 - 17

Many fast graph algorithms begin by preprocessing the graph to improve its sparsity. A common form of this is spectral sparsification, which involves removing...

positive semidefinite matrices | Laplacian matrix | derandomization | Spectral sparsifiers | randomized algorithms | Randomized algorithms | Positive semidefinite matrices | Derandomization | MATHEMATICS, APPLIED | SPECTRAL SPARSIFICATION | APPROXIMATIONS | PROGRAMS | GRAPH SPARSIFICATION | COMPUTER SCIENCE, THEORY & METHODS | ALGORITHMS | Algorithms | Tasks | Approximation | Preprocessing | Games | Graphs | Spectra | Sums

positive semidefinite matrices | Laplacian matrix | derandomization | Spectral sparsifiers | randomized algorithms | Randomized algorithms | Positive semidefinite matrices | Derandomization | MATHEMATICS, APPLIED | SPECTRAL SPARSIFICATION | APPROXIMATIONS | PROGRAMS | GRAPH SPARSIFICATION | COMPUTER SCIENCE, THEORY & METHODS | ALGORITHMS | Algorithms | Tasks | Approximation | Preprocessing | Games | Graphs | Spectra | Sums

Journal Article

Journal of Machine Learning Research, ISSN 1532-4435, 02/2011, Volume 12, pp. 593 - 625

The paper addresses the problem of learning a regression model parameterized by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear nature...

Riemannian geometry | Gradient-based learning | Positive semidefinite matrices | Low-rank approximation | Linear regression | positive semidefinite matrices | gradient-based learning | EXPONENTIATED GRADIENT | linear regression | ALGORITHMS | low-rank approximation | SUBSPACE | AUTOMATION & CONTROL SYSTEMS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | GEOMETRY

Riemannian geometry | Gradient-based learning | Positive semidefinite matrices | Low-rank approximation | Linear regression | positive semidefinite matrices | gradient-based learning | EXPONENTIATED GRADIENT | linear regression | ALGORITHMS | low-rank approximation | SUBSPACE | AUTOMATION & CONTROL SYSTEMS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | GEOMETRY

Journal Article

LINEAR ALGEBRA AND ITS APPLICATIONS, ISSN 0024-3795, 07/2019, Volume 572, pp. 51 - 67

It is obtained that for positive semidefinite block matrix H = [M K K* N] if A, B is an element of M-n (C) such that max(parallel to A parallel to(2), parallel...

MATHEMATICS | MATHEMATICS, APPLIED | Unitarily invariant norm | Majorization | Eigenvalue | MAJORIZATION INEQUALITIES | SINGULAR-VALUES | Convex function | Positive semidefinite matrix | Inequality

MATHEMATICS | MATHEMATICS, APPLIED | Unitarily invariant norm | Majorization | Eigenvalue | MAJORIZATION INEQUALITIES | SINGULAR-VALUES | Convex function | Positive semidefinite matrix | Inequality

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 01/2007, Volume 76, Issue 257, pp. 287 - 298

For the non-Hermitian and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the unconditional...

Linear systems | Economic theory | Iterative solutions | Eigenvalues | Matrices | Eigenvectors | Mathematics | Linear equations | Coefficients | Perceptron convergence procedure | Positive semidefinite matrix | Hermitian and skew-Hermitian splitting | Splitting iteration method | Convergence | Non-Hermitian matrix

Linear systems | Economic theory | Iterative solutions | Eigenvalues | Matrices | Eigenvectors | Mathematics | Linear equations | Coefficients | Perceptron convergence procedure | Positive semidefinite matrix | Hermitian and skew-Hermitian splitting | Splitting iteration method | Convergence | Non-Hermitian matrix

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 04/2014, Volume 260, pp. 236 - 243

In this paper, we consider the low rank approximation of the symmetric positive semidefinite matrix, which arises in machine learning, quantum chemistry and...

Nonlinear conjugate gradient method | Unconstrained optimization | Feasible set | Low rank approximation | Symmetric positive semidefinite matrix | MATHEMATICS, APPLIED | GCDS | Conjugate gradient method | Inverse problems | Approximation | Mathematical analysis | Transforms | Mathematical models | Optimization | Symmetry

Nonlinear conjugate gradient method | Unconstrained optimization | Feasible set | Low rank approximation | Symmetric positive semidefinite matrix | MATHEMATICS, APPLIED | GCDS | Conjugate gradient method | Inverse problems | Approximation | Mathematical analysis | Transforms | Mathematical models | Optimization | Symmetry

Journal Article

JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION, ISSN 1547-5816, 04/2019, Volume 15, Issue 2, pp. 893 - 908

A symmetric matrix A is completely positive (CP) if there exists an entrywise nonnegative matrix V such that A = VVT. A real symmetric matrix is called...

OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | completely positive separability | ENGINEERING, MULTIDISCIPLINARY | moment problems | ELLIPTICITY | semidefinite algorithm | POLYNOMIAL OPTIMIZATION | CONE | Completely positive matrices

OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | completely positive separability | ENGINEERING, MULTIDISCIPLINARY | moment problems | ELLIPTICITY | semidefinite algorithm | POLYNOMIAL OPTIMIZATION | CONE | Completely positive matrices

Journal Article

Discrete & Computational Geometry, ISSN 0179-5376, 10/2013, Volume 50, Issue 3, pp. 679 - 699

The positive semidefinite (psd) rank of a polytope is the smallest $$k$$ k for which the cone of $$k \times k$$ k × k real symmetric psd matrices admits an...

Polytope | Slack matrix | Cone lift | Computational Mathematics and Numerical Analysis | Positive semidefinite rank | Mathematics | Combinatorics | Hadamard square roots | GRAPH | MATHEMATICS | COMPUTER SCIENCE, THEORY & METHODS | NONNEGATIVE RANK | Polytopes | Matrix | Lower bounds | Computational geometry

Polytope | Slack matrix | Cone lift | Computational Mathematics and Numerical Analysis | Positive semidefinite rank | Mathematics | Combinatorics | Hadamard square roots | GRAPH | MATHEMATICS | COMPUTER SCIENCE, THEORY & METHODS | NONNEGATIVE RANK | Polytopes | Matrix | Lower bounds | Computational geometry

Journal Article

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