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 $\varepsilon$-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 $\varepsilon$ or certifies its $\varepsilon$-global optimality.

The framework is evaluated on pressure reconstruction for NACA airfoils using XFOIL data.