Matrix computations (3rd ed.)
A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation
ACM Transactions on Mathematical Software (TOMS)
Global Optimization with Polynomials and the Problem of Moments
SIAM Journal on Optimization
Real solving for positive dimensional systems
Journal of Symbolic Computation
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Numerical Polynomial Algebra
Smooth minimization of non-smooth functions
Mathematical Programming: Series A and B
Local Minima and Convergence in Low-Rank Semidefinite Programming
Mathematical Programming: Series A and B
Fast maximum margin matrix factorization for collaborative prediction
ICML '05 Proceedings of the 22nd international conference on Machine learning
On the solution of large-scale SDP problems by the modified barrier method using iterative solvers
Mathematical Programming: Series A and B
Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals
Foundations of Computational Mathematics
Thirty years of Polynomial System Solving, and now?
Journal of Symbolic Computation
Solving polynomial systems via symbolic-numeric reduction to geometric involutive form
Journal of Symbolic Computation
Subdivision methods for solving polynomial equations
Journal of Symbolic Computation
A prolongation-projection algorithm for computing the finite real variety of an ideal
Theoretical Computer Science
An accelerated gradient method for trace norm minimization
ICML '09 Proceedings of the 26th Annual International Conference on Machine Learning
Regularization Methods for Semidefinite Programming
SIAM Journal on Optimization
A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems
SIAM Journal on Imaging Sciences
A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems
SIAM Journal on Imaging Sciences
Interior-Point Method for Nuclear Norm Approximation with Application to System Identification
SIAM Journal on Matrix Analysis and Applications
Optimization Methods & Software
A Newton-CG Augmented Lagrangian Method for Semidefinite Programming
SIAM Journal on Optimization
Convergence of Fixed-Point Continuation Algorithms for Matrix Rank Minimization
Foundations of Computational Mathematics
The minimum-rank gram matrix completion via modified fixed point continuation method
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Verified error bounds for real solutions of positive-dimensional polynomial systems
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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In this paper, we propose a new algorithm for computing real roots of polynomial equations or a subset of real roots in a given semi-algebraic set described by additional polynomial inequalities. The algorithm is based on using modified fixed point continuation method for solving Lasserre's hierarchy of moment relaxations. We establish convergence properties for our algorithm. For a large-scale polynomial system with only few real solutions in a given area, we can extract them quickly. Moreover, for a polynomial system with an infinite number of real solutions, our algorithm can also be used to find some isolated real solutions or real solutions on the manifolds.