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.