A Max-Plus-Based Algorithm for a Hamilton--Jacobi--Bellman Equation of Nonlinear Filtering
SIAM Journal on Control and Optimization
Global Optimization with Polynomials and the Problem of Moments
SIAM Journal on Optimization
Journal of Global Optimization
SIAM Journal on Optimization
A Curse-of-Dimensionality-Free Numerical Method for Solution of Certain HJB PDEs
SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
ACM Transactions on Mathematical Software (TOMS)
Computing sum of squares decompositions with rational coefficients
Theoretical Computer Science
MetiTarski: An Automatic Theorem Prover for Real-Valued Special Functions
Journal of Automated Reasoning
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
Chebyshev interpolation polynomial-based tools for rigorous computing
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Mathematical Programming: Series A and B
On the generation of positivstellensatz witnesses in degenerate cases
ITP'11 Proceedings of the Second international conference on Interactive theorem proving
Journal of Symbolic Computation
Positivity and Optimization for Semi-Algebraic Functions
SIAM Journal on Optimization
Scalable analysis of linear systems using mathematical programming
VMCAI'05 Proceedings of the 6th international conference on Verification, Model Checking, and Abstract Interpretation
Hi-index | 0.00 |
The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of inequalities. We introduce an approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis). This algorithm consists in bounding some of the constituents of the function by suprema of quadratic forms with a well chosen curvature. This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation. We illustrate the efficiency of our framework with various examples from the literature and discuss the interfacing with Coq.