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Modeling, Analysis and Simulation: Global Optimization of Rational Functions: a Semidefinite Programming Approach

机译：建模，分析和模拟：有理函数的全局优化：半定规划方法

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We consider the problem of global minimization of rational functions on IRn (unconstrained case), and on an open, connected, semi-algebraic subset of IRn, or the (partial) closure of such a set (constrained case). We show that in the univariate case (n = 1), these problems have exact reformulations as semidefinite programming (SDP) problems, by using reformulations introduced in the PhD thesis of Jibetean. This extends the analogous results by Nesterov for global minimization of univariate polynomials. For the bivariate case (n = 2), we obtain a fully polynomial time approximation scheme (FPTAS) for the unconstrained problem, if an a priori lower bound on the infimum is known, by using results by De Klerk and Pasechnik. For the NP-hard multivariate case, we discuss semidefinite programmingbased relaxations for obtaining lower bounds on the infimum, by using results by Parrilo, and Lasserre.

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