Certification · Semidefinite Programming · Transcendental Functions · Semialgebraic

Abstract We consider the problem of certifying lower bounds for real-valued mul-tivariate transcendental functions. The functions we are dealing with are nonlinear and involve semialgebraic operations as well as some transcendental functions like cos, arctan, exp, etc. Our general framework is to use different approximation meth-ods to relax the original problem into polynomial optimization problems, which we solve by sparse sums of squares relaxations. In particular, we combine the ideas of the maxplus estimators (originally introduced in optimal control) and of the linear templates (originally introduced in static analysis by abstract interpretation). The nonlinear templates control the complexity of the semialgebraic relaxations at the price of coarsening the maxplus approximations. In that way, we arrive at a new-template based- certified global optimization method, which exploits both the pre-cision of sums of squares relaxations and the scalability of abstraction methods. We analyze the performance of the method on problems from the global optimization literature, as well as medium-size inequalities issued from the Flyspeck project