TY - JOUR

T1 - Global optimization of rational functions : a semidefinite programming approach

AU - Jibetean, D.

AU - Klerk, de, E.

PY - 2006

Y1 - 2006

N2 - 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].

AB - 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].

U2 - 10.1007/s10107-005-0589-0

DO - 10.1007/s10107-005-0589-0

M3 - Article

VL - 106

SP - 93

EP - 109

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1

ER -