Regularization Methods for Sum of Squares Relaxations in Large Scale Polynomial Optimization , preprint, 2009.
Polynomial Matrix Inequality and Semidefinite Representation, preprint, 2009.
An Approximation Bound Analysis for Lasserre's Relaxation in Multivariate Polynomial Optimization , preprint, 2009.
Sum of Squares Methods for Minimizing Polynomial Forms over Spheres and Hypersurfaces , preprint, 2009.
First Order Conditions for Semidefinite Representations of Convex Sets Defined by
Rational or Singular Polynomials, preprint, 2008.
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.
Structured Semidefinite Representation of Some Convex Sets
Joint with J.W. Helton, Proceedings of 47th IEEE Conference on Decision and Control, pp. 4797 - 4800, Cancun, Mexico, Dec. 9-11, 2008.
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).
Solving partial differential equations via sparse SDP relaxations
Joint with M. Mevissen, M. Kojima, and N. Takayama, Pacific Journal of Optimization, Vol. 4 (2) 213 - 241 (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.
An elementary and constructive proof of Hilbert's 17th Problem for matrices
Joint with C.Hillar, Proceedings of the American Mathematical Society, 136 (2008), 73-76.
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 Complexity 23(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.
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.