Global optimization of rational functions: a semidefinite programming approach

Authors:
D. Jibetean;E. de Klerk
Affiliations:
Dept. Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600, Eindhoven, MB, The Netherlands;Department of Econometrics and Operations Research, Tilburg University, P.O. Box 90153, 5000, Tilburg, LE, The Netherlands
Venue:
Mathematical Programming: Series A and B
Year:
2006

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Abstract

We consider the problem of global minimization of rational functions on ** (unconstrained case), and on an open, connected, semi-algebraic subset of **, 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 [16]. This extends the analogous results by Nesterov [13] 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 [1].For the NP-hard multivariate case, we discuss semidefinite programming-based relaxations for obtaining lower bounds on the infimum, by using results by Parrilo [15], and Lasserre [12].