SIAM Review
On the combinatorial and algebraic complexity of quantifier elimination
Journal of the ACM (JACM)
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
Convex Optimization
On the complexity of Schmüdgen's positivstellensatz
Journal of Complexity
Signature of symmetric rational matrices and the unitary dual of lie groups
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Implementing the cylindrical algebraic decomposition within the Coq system
Mathematical Structures in Computer Science
Algorithm 875: DSDP5—software for semidefinite programming
ACM Transactions on Mathematical Software (TOMS)
Computing the global optimum of a multivariate polynomial over the reals
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Computing sum of squares decompositions with rational coefficients
Theoretical Computer Science
Fast reflexive arithmetic tactics the linear case and beyond
TYPES'06 Proceedings of the 2006 international conference on Types for proofs and programs
Verifying nonlinear real formulas via sums of squares
TPHOLs'07 Proceedings of the 20th international conference on Theorem proving in higher order logics
Journal of Symbolic Computation
An algebraic approach for the unsatisfiability of nonlinear constraints
CSL'05 Proceedings of the 19th international conference on Computer Science Logic
Formalization of Bernstein Polynomials and Applications to Global Optimization
Journal of Automated Reasoning
Certification of bounds of non-linear functions: the templates method
CICM'13 Proceedings of the 2013 international conference on Intelligent Computer Mathematics
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One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some linear equality (Positivstellen-satz). This produces a witness for the desired property, from which it is reasonably easy to obtain a formal proof of the property suitable for a proof assistant such as Coq. The problem of finding a witness reduces to a feasibility problem in semidefinite programming, for which there exist numerical solvers. Unfortunately, this problem is in general not strictly feasible, meaning the solution can be a convex set with empty interior, in which case the numerical optimization method fails. Previously published methods thus assumed strict feasibility; we propose a workaround for this difficulty. We implemented our method and illustrate its use with examples, including extractions of proofs to Coq.