Semidefinite Approximations for Global Unconstrained Polynomial Optimization

A semidefinite relaxation scheme for quadratically constrained

  Semidefinite optimization relaxations are among the widely used approaches to find global optimal or approximate solutions for many nonconvex problems. Here, we consider a specific quadratically constrained quadratic problem with an additional linear constraint. We prove that under certain conditions the semidefinite relaxation approach enables us to find a global optimal solution of the unde...

Global optimization of rational functions: a semidefinite programming approach

We consider the problem of global minimization of rational functions on IR (unconstrained case), and on an open, connected, semi-algebraic subset of IR, 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...

Converging Semidefinite Bounds for Global Unconstrained Polynomial Optimization

We consider here the problem of minimizing a polynomial function on Rn. The problem is known to be hard even for degree 4. Therefore approximation algorithms are of interest. Lasserre [11] and Parrilo [16] have proposed approximating the minimum of the original problem using a hierarchy of lower bounds obtained via semidefinite programming relaxations. We propose here a method for computing a c...

New complexity analysis of a full Nesterov-Todd steps IIPM for semidefinite optimization

A Semidefinite Optimization Approach to Quadratic Fractional Optimization with a Strictly Convex Quadratic Constraint

In this paper we consider a fractional optimization problem that minimizes the ratio of two quadratic functions subject to a strictly convex quadratic constraint. First using the extension of Charnes-Cooper transformation, an equivalent homogenized quadratic reformulation of the problem is given. Then we show that under certain assumptions, it can be solved to global optimality using semidefini...