  This paper focuses on designing edge-weighted networks, whose robustness is
characterized by maximizing algebraic connectivity, or the second smallest
eigenvalue of the Laplacian matrix. This problem is motivated by cooperative
vehicle localization, where accurately estimating relative position
measurements and establishing communication links are essential. We also
examine an associated problem where every robot is limited by payload, budget,
and communication to pick no more than a specified number of relative position
measurements. The basic underlying formulation for these problems is nonlinear
and is known to be NP-hard. Our approach formulates this problem as a Mixed
Integer Semi-Definite Program (MISDP), later reformulated into a Mixed Integer
Linear Program (MILP) for obtaining optimal solutions using cutting plane
algorithms. We introduce a novel upper-bounding algorithm based on principal
minor characterization of positive semi-definite matrices and discuss a
degree-constrained lower bounding formulation inspired by robust network
structures. In addition, we propose a maximum cost heuristic with low
computational complexity to identify high-quality feasible solutions for
instances involving up to one hundred nodes. We show extensive computational
results corroborating our proposed methods.
