On the complexity of Putinar's Positivstellensatz
Journal of Complexity
Global minimization of rational functions and the nearest GCDs
Journal of Global Optimization
Computing the global optimum of a multivariate polynomial over the reals
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Sum of squares method for sensor network localization
Computational Optimization and Applications
Global optimization of polynomials using generalized critical values and sums of squares
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Biquadratic Optimization Over Unit Spheres and Semidefinite Programming Relaxations
SIAM Journal on Optimization
Deciding reachability of the infimum of a multivariate polynomial
Proceedings of the 36th international symposium on Symbolic and algebraic computation
An iterative scheme for valid polynomial inequality generation in binary polynomial programming
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Discriminants and nonnegative polynomials
Journal of Symbolic Computation
Representations of Positive Polynomials and Optimization on Noncompact Semialgebraic Sets
SIAM Journal on Optimization
SIAM Journal on Optimization
Second-Order Cone Relaxations for Binary Quadratic Polynomial Programs
SIAM Journal on Optimization
Global optimization of polynomials restricted to a smooth variety using sums of squares
Journal of Symbolic Computation
Exploiting equalities in polynomial programming
Operations Research Letters
Minimizing rational functions by exact Jacobian SDP relaxation applicable to finite singularities
Journal of Global Optimization
Lower bounds on the global minimum of a polynomial
Computational Optimization and Applications
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A method is proposed for finding the global minimum of a multivariate polynomial via sum of squares (SOS) relaxation over its gradient variety. That variety consists of all points where the gradient is zero and it need not be finite. A polynomial which is nonnegative on its gradient variety is shown to be SOS modulo its gradient ideal, provided the gradient ideal is radical or the polynomial is strictly positive on the real gradient variety. This opens up the possibility of solving previously intractable polynomial optimization problems. The related problem of constrained minimization is also considered, and numerical examples are discussed. Experiments show that our method using the gradient variety outperforms prior SOS methods.