Book Chapters

A complete semidefinite algorithm for detecting copositive matrices and tensors

Joint with Z. Yang, X. Zhang,SIAM Journal on Optimization28(4), 2902-2921, 2018.

Real eigenvalues of nonsymmetric tensors

Joint with X. Zhang,Computational Optimization and Applications70(1), 1-32, 2018.

Tensor Eigenvalue Complementarity Problems

Joint with J. Fan, A. Zhou,Mathematical Pogramming, Ser. A, 170(2): 507-539, 2018.

Low Rank Symmetric Tensor Approximations

SIAM Journal on Matrix Analysis and Applications, Vol. 38, No. 4, pp. 1517-1540, 2017.

Symmetric Tensor Nuclear Norms

SIAM Journal on Applied Algebra and Geometry, Vol. 1, No. 1, pp. 599-625, 2017.

Bilevel Polynomial Programs and Semidefinite Relaxation Methods

Joint with L. Wang, J. Ye,SIAM Journal on Optimization, Vol. 27, No. 3, pp. 1728-1757, 2017.

Generating Polynomials and Symmetric Tensor Decompositions

Foundations of Computational Mathematics, Vol. 17, No. 2, pp. 423-465, 2017.

Positive Maps and Separable Matrices

Joint with X. Zhang,SIAM Journal on Optimization, Vol. 26, No. 2, pp. 1236-1256, 2016

Linear Optimization with Cones of Moments and Nonnegative Polynomials

Mathematical Programming, Ser. B, 153(1): 247-274, 2015.

The Hierarchy of Local Minimums in Polynomial Optimization

Mathematical Programming, Ser. B,Vol. 151, No.2, pp. 555-583, 2015.

All Real Eigenvalues of Symmetric Tensors.

Joint with C. Cui and Y. Dai.SIAM Jounral on Matrix Analysis and Applications, Vol. 35, No. 4, pp. 1582-1601, 2014.

(The Matlab code is here).

Semidefinite Relaxations for Best Rank-1 Tensor Approximations

Joint with L. Wang.SIAM Jounral on Matrix Analysis and Applications, Vol. 35, No. 3, pp. 1155-1179, 2014.

(The Matlab code is here).

The A-Truncated K-Moment Problem

Foundations of Computational Mathematics, Vol. 14, No. 6, pp. 1243-1276, 2014.

Optimality Conditions and Finite Convergence of Lasserre's Hierarchy

Mathematical Programming, Ser. A,Vol 146, No. 1-2, pp. 97-121, 2014.

Polynomial Optimization with Real Varieties

SIAM Journal on Optimization, Vol 23, No.3, pp. 1634-1646, 2013.

Certifying Convergence of Lasserre's Hierarchy via Flat Truncation

Mathematical Programming, Ser. A,Vol 142, No. 1-2, pp. 485-510, 2013.

An Approximation Bound Analysis for Lasserre's Relaxation in Multivariate Polynomial Optimization

Journal of the Operations Research Society of China, Vol. 1, No. 3, pp. 313-332, 2013.

A Semidefinite Approach for Truncated K-Moment Problem

Joint with J.W. Helton,Foundations of Computational Mathematics,Vol. 12, No. 6, pp. 851-881, 2012.

An Exact Jacobian SDP Relaxation for Polynomial Optimization

Mathematical Programming, Ser. A,Vol. 137, No. 1-2, pp. 225-255, 2013.

The truncated moment problem via homogenization and flat extensions

Joint with L. Fialkow,Journal of Functional Analysis263 (2012), no. 6, 1682-1700.

Regularization Methods for SDP Relaxations in Large Scale Polynomial Optimization

Joint with L. Wang,SIAM Journal On Optimization,Vol. 22, No. 2, pp. 408-428, 2012. (The Matlab code is here).

First Order Conditions for Semidefinite Representations of Convex Sets Defined by Rational or Singular Polynomials

Mathematical Programming, Ser. A,Vol. 131, No. 1, pp. 1-36, 2012.

Discriminants and Nonnegative Polynomials

Journal of Symbolic Computation,Vol. 47, No. 2, pp. 167-191, 2012.

Sum of squares methods for minimizing polynomial functions over spheres and hypersurfaces,

Frontiers of Mathematics in China,Vol 7, No.2, pp. 321-346, 2012.

Polynomial Matrix Inequality and Semidefinite Representation

Mathematics of Operations Research,Vol. 36, No. 3, pp. 398-415, 2011.

Positivity of Riesz Functionals and Solutions of Quadratic and Quartic Moment Problems

Joint with L. Fialkow,Journal of Functional Analysis, Vol. 258, No. 1, pp. 328-356, 2010.

The Algebraic Degree of Semidefinite Programming

Joint with K. Ranestad and B. Sturmfels,Mathematical Programming, Series A, Vol. 122, No.2, pp. 379-405, 2010.

Semidefinite Representation of Convex Sets

Joint with J.W. Helton,Mathematical Programming, Series A,Vol. 122, No.1, pp.21-64, 2010.

Bi-Quadratic Optimization over Unit Spheres and Semidefinite Programming Relaxations

Joint with C. Ling, L. Qi, and Y. Ye,SIAM Journal on Optimization, Vol. 20, No. 3, pp.1286-1310, 2009.

Matrix cubes parametrized by eigenvalues

Joint with B. Sturmfels,SIAM Journal on Matrix Analysis and Applications, Vol. 31, No. 2, pp. 755-766, 2009.

Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets

Joint with J.W. Helton,SIAM Journal on Optimization, Vol. 20, No.2, pp. 759-791, 2009.

Sum of squares method for sensor network localization

Computational Optimization and Applications, Vol.43, No. 2 (2009), pp. 151-179.

Algebraic Degree of Polynomial Optimization

Joint with K. Ranestad,SIAM Journal on Optimization, Vol. 20, No. 1, pp. 485-502, 2009.

Sparse SOS relaxations for minimizing functions that are summations of small polynomials

Join with J. Demmel,SIAM Journal On Optimization, Vol. 19, No. 4, pp. 1534-1558 (2008).

Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization

Joint with S. He, Z. Luo, and S. Zhang,SIAM Journal on Optimization, Vol. 19, No.2, pp. 503-523, 2008.

Global minimization of rational functions and the nearest GCDs

Joint with J. Demmel and M. Gu,Journal of Global Optimization, Vol. 40 (2008), No.4, 697-718.

On the complexity of putinar's positivstellensatz

Joint with M. Schweighofer,Journal of Complexity23(2007), pp.135-150.

Representations of positive polynomials on non-compact semialgebraic sets via KKT ideals

Joint with J. Demmel and V. Powers,Journal of Pure and Applied Algebra, Vol. 209, No. 1, pp. 189-200, 2007.

Minimizing polynomials via sum of squares over the gradient ideal

Joint with J. Demmel and B. Sturmfels,Mathematical Programming, Series A, Vol. 106 (2006), No. 3, pp. 587-606.

Minimum ellipsoid bounds for solutions of polynomial systems via sum of squares

Joint with J. Demmel,Journal of Global Optimization, Vol. 33(2005), pp. 511-525.

Convex Hulls of Quadratically Parameterized Sets With Quadratic Constraints

Mathematical Methods in Systems, Optimization and Control(eds. Harry Dym, Mauricio de Oliveira, Mihai Putinar). Operator Theory: Advances and Applications, Vol 222, pp. 247-258, Birkhauser, Basel, 2012.

Approximate GCDs of polynomials and sparse SOS relaxations

Joint with B. Li and L. Zhi,Theoretical Computer Science, 409(2) pp.200-210, 2008.

Semidefinite Representation of the k-Ellipse

Joint with P. Parrilo and B. Sturmfels,IMA Volume 146: Algorithms in Algebraic Geometry

(Eds. A. Dickenstein, F.-O. Schreyer, and A. Sommese), pp. 117-132, Springer, New York, 2008.

Shape optimization of transfer functions

Joint with J. Demmel,Multiscale optimization methods and applications, pp. 313--326,

Nonconvex Optim. Appl., 82, Springer, New York, 2006.