Approximate maximum-likelihood estimation using semidefinite programming

We consider semidefinite relaxations of a quadratic optimization problem with polynomial constraints. This is an extension of quadratic problems with boolean variables. Such combinatorial problems can in general not be solved in polynomial time. Semidefinite relaxations has been proposed as a promising technique to give provable good bounds on certain boolean quadratic problems in polynomial time. We formulate the extensions from boolean variables to quarternary variables using (i) a polynomial relaxation or (ii) by using standard semidefinite relaxations of a linear transformation of boolean variables. We analytically compare the two different approaches of relaxation. The relaxations can all be expressed as semidefinite programs, which can be solved efficiently using e.g. interior point methods. Applications of our results include maximum likelihood estimation in communication systems, which we explore in simulations in order to compare the quality of the different relaxations with optimal solutions.