Global solutions for the sensors placement problem via weakly convex optimization

We address the problem of optimally placing a limited number of sensors to reconstruct high-dimensional signals without knowledge of the underlying dynamics. The task is formulated as a nonconvex combinatorial optimisation problem and recast as a weakly convex constrained projection problem. This reformulation allows us to compute $\epsilon$-global solutions using the Inexact Cutting Sphere algorithm. We further propose the Inverse Cutting Sphere algorithm, which starts from any feasible heuristic solution and either improves it by a prescribed tolerance $\epsilon$ or certifies its $\epsilon$-global optimality. The framework is evaluated on pressure reconstruction for NACA airfoils using XFOIL data.