Global Optimization with Polynomials and the Problem of Moments
SIAM Journal on Optimization
Minimizing Polynomials via Sum of Squares over the Gradient Ideal
Mathematical Programming: Series A and B
GloptiPoly 3: moments, optimization and semidefinite programming
Optimization Methods & Software - GLOBAL OPTIMIZATION
The algebraic degree of semidefinite programming
Mathematical Programming: Series A and B
Algebraic Degree of Polynomial Optimization
SIAM Journal on Optimization
Matrix Cubes Parameterized by Eigenvalues
SIAM Journal on Matrix Analysis and Applications
Biquadratic Optimization Over Unit Spheres and Semidefinite Programming Relaxations
SIAM Journal on Optimization
Minimizing rational functions by exact Jacobian SDP relaxation applicable to finite singularities
Journal of Global Optimization
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For a semialgebraic set K in R^n, let P"d(K)={f@?R[x]"@?"d:f(u)=0@?u@?K} be the cone of polynomials in x@?R^n of degrees at most d that are nonnegative on K. This paper studies the geometry of its boundary @?P"d(K). We show that when K=R^n and d is even, its boundary @?P"d(K) lies on the irreducible hypersurface defined by the discriminant @D(f) of f. We show that when K={x@?R^n:g"1(x)=...=g"m(x)=0} is a real algebraic variety, @?P"d(K) lies on the hypersurface defined by the discriminant @D(f,g"1,...,g"m) of f,g"1,...,g"m. We show that when K is a general semialgebraic set, @?P"d(K) lies on a union of hypersurfaces defined by the discriminantal equations. Explicit formulae for the degrees of these hypersurfaces and discriminants are given. We also prove that typically P"d(K) does not have a barrier of type -log@f(f) when @f(f) is required to be a polynomial, but such a barrier exists if @f(f) is allowed to be semialgebraic. Some illustrating examples are shown.