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

TL;DR

This paper introduces a semidefinite programming method to minimize ordered weighted averages of rational functions over semi-algebraic sets, enabling solutions to complex location problems in higher dimensions.

Contribution

It develops a novel SDP-based reformulation and hierarchy of relaxations for minimizing ordered weighted averages of rational functions, extending solvable problem classes.

Findings

Hierarchy of SDP relaxations approximates optimal values arbitrarily closely.

Method applies to convex and non-convex location problems in any finite dimension.

Extensive computational results demonstrate effectiveness in 2D and 3D location problems.

Abstract

This paper considers the problem of minimizing the ordered weighted average (or ordered median) function of finitely many rational functions over compact semi-algebraic sets. Ordered weighted averages of rational functions are not, in general, neither rational functions nor the supremum of rational functions so that current results available for the minimization of rational functions cannot be applied to handle these problems. We prove that the problem can be transformed into a new problem embedded in a higher dimension space where it admits a convenient representation. This reformulation admits a hierarchy of SDP relaxations that approximates, up to any degree of accuracy, the optimal value of those problems. We apply this general framework to a broad family of continuous location problems showing that some difficult problems (convex and non-convex) that up to date could only be solved on the plane and with Euclidean distance, can be reasonably solved with different $\ell_p$-norms and in any finite dimension space. We illustrate this methodology with some extensive computational results on location problems in the plane and the 3-dimension space.