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This paper focuses on the design of edge-weighted networks, whose robustnessis characterized by maximizing algebraic connectivity, or the smallest non-zeroeigenvalue of the Laplacian matrix. This problem is motivated by theapplication of cooperative localization for accurately estimating positions ofautonomous vehicles by choosing a set of relative position measurements andestablishing associated communication links. We also examine an associatedproblem where every robot is limited by payload, budget, and communication topick no more than a specified number of relative position measurements. Thebasic underlying formulation for these problems is nonlinear and is known to beNP-hard. We solve this network design problem by formulating it as amixed-integer semi-definite program (MISDP) and reformulating it into amixed-integer linear program to obtain optimal solutions using cutting planealgorithms. We propose a novel upper-bounding algorithm based on the hierarchyof principal minor characterization of positive semi-definite matrices. Wefurther discuss a degree-constrained lower bounding formulation, inspired byrobust network structures. In addition, we propose a maximum-cost heuristicwith low computational complexity to find high-quality feasible solutions. Weshow extensive computational results corroborating our proposed methods.

Optimal Robust Network Design: Formulations and Algorithms for Maximizing Algebraic Connectivity