Polar varieties, real equation solving, and data structures: the hypersurface case
Journal of Complexity
A Gröbner free alternative for polynomial system solving
Journal of Complexity
Real solving for positive dimensional systems
Journal of Symbolic Computation
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Minimizing Polynomials via Sum of Squares over the Gradient Ideal
Mathematical Programming: Series A and B
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Computing the global optimum of a multivariate polynomial over the reals
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Computing sum of squares decompositions with rational coefficients
Theoretical Computer Science
The Voronoi Diagram of Three Lines
Discrete & Computational Geometry - 23rd Annual Symposium on Computational Geometry
Proceedings of the 2009 conference on Symbolic numeric computation
On the intrinsic complexity of point finding in real singular hypersurfaces
Information Processing Letters
On the geometry of polar varieties
Applicable Algebra in Engineering, Communication and Computing
Global optimization of polynomials using generalized critical values and sums of squares
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Journal of Symbolic Computation
Computing Rational Points in Convex Semialgebraic Sets and Sum of Squares Decompositions
SIAM Journal on Optimization
Computing rational solutions of linear matrix inequalities
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
Verified error bounds for real solutions of positive-dimensional polynomial systems
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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Let f ∈ Q[X_1,...,Xn] be of degree D. Algorithms for solving the unconstrained global optimization problem f*=infx∈Rn f(x) are of first importance since this problem appears frequently in numerous applications in engineering sciences. This can be tackled by either designing appropriaten quantifier elimination algorithms or by certifying lower bounds on f* by means of sums of squares decompositions but there is no efficient algorithm for deciding if f* is a minimum. This paper is dedicated to this important problem. We design a probabilistic algorithm that decides, for a given f and the corresponding f*, if f* is reached over Rn and computes a point x* ∈ Rn such that f(x*)=f* if such a point exists. This algorithm makes use of algebraic elimination algorithms and real root isolation. If L is the length of a straight-line program evaluating f, algebraic elimination steps run in O(log(D-1)n6(nL+n4)U((D-1)n+1)3) arithmetic operations in Q where D=deg(f) and U(x)=x(log (x))2 log log (x). Experiments show its practical efficiency.