Accelerating energy minimization process in charged polymer–biomolecular systems: An enhanced nonlinear conjugate gradient method

Energy minimization in charged polymer-multi-biomolecule electrolyte solution systems faces major challenges, where the energy landscape is typically highly nonconvex, ill-conditioned, and dominated by long-range electrostatic interactions. In such settings, standard nonlinear conjugate gradient (NCG) methods often struggle to maintain sufficient descent directions due to unstable conjugate gradient parameters caused by poor curvature information and frequent oscillations in gradient directions. To address this, we develop an enhanced NCG algorithm, termed the ELS (short for Enlong Shang) method, which introduces a modified conjugate gradient coefficient βkELS with a tunable denominator parameter ω, enabling improved stability in regions with poor local curvature. In addition, existing studies have shown that the convergence analysis of current NCG methods usually relies on the pre-setting of the parameter σ, whose theoretical bounds are difficult to adapt to the complex demands of high-dimensional nonconvex optimization problems. Hence, a novel convergence proof technique is proposed to show that the ELS method satisfies the sufficient descent condition for a broad range of line search parameters σ ∈ (0, 1), while still ensuring global convergence under nonconvex objectives. For traditional unconstrained optimization problems, the numerical performance of the ELS method outperforms the existing representative NCG methods. We apply it to the energy minimization phase in complex biomolecular simulations. Compared to direct dynamics simulation without preprocessing, implementing this minimization saves about 60% of the total time required to reach dynamic equilibrium, even exceeding the mainstream staged minimization strategy in Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). Importantly, the final conformation closely matches that of the purely dynamics simulation thermodynamically and has an acceptable energy deviation.