Global optimization of rational functions:A semidefinite programming approach

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]

An algebraic method for system reduction of stationary Gaussian systems

System identification for a particular approach reduces to system reduction, determining for a system with a high state-space dimension a system of low state-space dimension. For Gaussian systems the problem of system reduction is considered with the divergence rate criterion. The divergence or Kullback-Leibler pseudo-distance corresponds to the expected value of the negative natural logarithm of the likelihood function. System reduction for Gaussian systems is thus a certainty equivalent way of maximum likelihood identification. An algebraic method is proposed for system reduction. The results are a theorem that this problem reduces to an infimization problem for a rational function for which programs are available and a procedure for computing the best approximant w.r.t. the divergence rate criterion. As illustration two examples of system reduction are presented

Global optimization of rational functions : a semidefinite programming approach

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]

Global Optimization of Rational Functions: A Semidefinite Programming Approach

We consider the problem of global minimization of rational functions (unconstrained case), and on an open, connected, semi-algebraic subset of IR , or the (partial) closure of such a set (constrained case)

Global minimization of a multivariate polynomial using matrix methods

minimization of a mutivariate polynomial using matrix method