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<p>We consider a special nonconvex minimization problem with a sphere constraint, which arises as the discretization of the energy functional minimization problem of certain nonlinear Schrödinger equations. We establish the connection between the problem’s global minimizer and the eigenvector of a nonlinear eigenvalue problem (NEP), proving that the NEP admits a unique positive eigenvector. As a result, the minimization problem has a unique positive global minimizer. Based on these, we briefly discuss the convergence of several algorithms for this optimization problem. We then apply the Newton–Noda iteration and the projected gradient method to find the global minimizer of the problem. The numerical results suggest that the projected gradient method is more efficient for large size problems in two and three-dimensional cases.</p>

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Characterizing the Global Optimum of a Class of Nonconvex Optimization Problems with a Comparison of Several Algorithms

  • 期刊论文

  • 发表日期：

    2025年7月24日

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  • Yunfei Tang,
  • Qingzhi Yang

摘要

We consider a special nonconvex minimization problem with a sphere constraint, which arises as the discretization of the energy functional minimization problem of certain nonlinear Schrödinger equations. We establish the connection between the problem’s global minimizer and the eigenvector of a nonlinear eigenvalue problem (NEP), proving that the NEP admits a unique positive eigenvector. As a result, the minimization problem has a unique positive global minimizer. Based on these, we briefly discuss the convergence of several algorithms for this optimization problem. We then apply the Newton–Noda iteration and the projected gradient method to find the global minimizer of the problem. The numerical results suggest that the projected gradient method is more efficient for large size problems in two and three-dimensional cases.