Recent Preprints
Journal Publications
Book Chapters

Recent Preprints

  • Certifying Convergence of Lasserre's Hierarchy via Flat Truncation , preprint, 2011.

  • A Semidefinite Approach for Truncated K-Moment Problem (joint with Bill Helton), preprint, 2011.

  • An Approximation Bound Analysis for Lasserre's Relaxation in Multivariate Polynomial Optimization , preprint, 2011.

    • Journal Publications

  • Regularization Methods for SDP Relaxations in Large Scale Polynomial Optimization
    Joint with L. Wang, SIAM Journal On Optimization, to appear. (The Matlab code is here).

  • An Exact Jacobian SDP Relaxation for Polynomial Optimization
    Mathematical Programming, Ser. A, to appear.

  • 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.

  • 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.

  • 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.

    • Book Chapters

  • Convex Hulls of Quadratically Parameterized Sets With Quadratic Constraints
    To appear in the volume in honor of Bill Helton, Mathematical Methods in Systems, Optimization and Control (eds. Harry Dym, Mauricio de Oliveira, Mihai Putinar).

  • 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.