A sphere packing approach to multi-parametric optimization

This paper introduces a novel application of sphere packing theory to multi-parametric optimization problems. Sphere packing, a classical problem in geometry, provides a powerful geometric framework for analyzing the robustness and feasibility of solutions in linear programming, inverse optimization, and multi-objective optimization problems. By inscribing the largest possible spheres within feasible regions, we develop robust solutions that can tolerate small perturbations in constraints or objectives. We derive theoretical results connecting sphere packing to linear programming, formulate robust solutions as ε-solutions, and address inverse optimization problems by maximizing the robustness of target solutions. A practical case study involving a mining company demonstrates the applicability of the proposed framework in real-world multi-objective scenarios. This approach enhances traditional optimization techniques by incorporating geometric insights, scalability, and robustness.