Beam Optics Ramping in Under-Constrained Lattice Design: Application to Electron-Ion Collider Hadron Storage Ring Cooling Section

This paper presents the lattice design and optics ramping strategy for the cooling section of the Hadron Storage Ring (HSR) at the Electron-Ion Collider (EIC). The main challenge is that available tuning knobs exceed beam-optics constraints. Independently optimized injection and top-energy optics often yield disconnected solutions, making interpolation impossible. To address this, we propose two new methods. The first is a midpoint-penalty scheme that ensures ramping path continuity by penalizing constraint violations at intermediate state. The second is a top-down approach that adapts high-energy optics to low energy, guided by an adaptive weighting scheme to balance injection and ramping constraints. The solutions meet all beam dynamics and hardware limits. The two methods offer a general strategy for ramping in systems where the solution space is under-constrained and the starting and target configurations are far apart.