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This paper considers the problem of minimizing the ordered weighted average(or ordered median) function of finitely many rational functions over compactsemi-algebraic sets. Ordered weighted averages of rational functions are not,in general, neither rational functions nor the supremum of rational functionsso that current results available for the minimization of rational functionscannot be applied to handle these problems. We prove that the problem can betransformed into a new problem embedded in a higher dimension space where itadmits a convenient representation. This reformulation admits a hierarchy ofSDP relaxations that approximates, up to any degree of accuracy, the optimalvalue of those problems. We apply this general framework to a broad family ofcontinuous location problems showing that some difficult problems (convex andnon-convex) that up to date could only be solved on the plane and withEuclidean distance, can be reasonably solved with different $\ell_p$-norms andin any finite dimension space. We illustrate this methodology with someextensive computational results on location problems in the plane and the3-dimension space.

A Semidefinite Programming approach for minimizing ordered weighted averages of rational functions