This paper investigates computational methods in quantum physics for approximating the ground state of Bose–Einstein condensates (BECs) by designing two relaxed formulations of the Gross–Pitaevskii energy functional. These formulations achieve first- and second-order accuracy with respect to the relaxation parameter \( \tau \) and are shown to converge to the original energy functional as \( \tau \rightarrow 0 \) . A key feature of the proposed relaxed models is their concavity, which ensures that local minima lie on the boundary of the concave hull. This property prevents energy increases during constraint normalization and enables the development of energy-dissipative algorithms. Numerical methods based on sequential linear programming are proposed, and their stability with respect to the relaxed energy is analyzed. To enhance computational efficiency, an adaptive strategy is introduced, dynamically refining solutions obtained with larger relaxation parameters to achieve higher accuracy with smaller ones. Numerical experiments demonstrate the stability, convergence, and energy dissipation of the proposed methods, while also demonstrating the effectiveness of the adaptive strategy in improving computational performance.
