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<p>This article considers optimization problems under non-linear partial differential equation (p.d.e.) constraints. It is assumed that the p.d.e. arises from minimizing a convex energy. We prove that the optimization problem is self-adjoint when the objective function is the dual energy. In other words, the differential of the objective function with respect to the optimization variable does not involve any adjoint state. This result generalizes the well-known fact that the so-called compliance is self-adjoint in the linear case. We also prove that in a large class of objective functions the dual energy is the only one which is self-adjoint.</p>

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A remark on self-adjoint problems in the optimization of non-linear models

  • 期刊论文

  • 发表日期：

    2026年2月18日

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  • Grégoire Allaire,
  • Théodore Cherrière,
  • Thomas Gauthey,
  • Maya Hage Hassan,
  • Xavier Mininger

摘要

This article considers optimization problems under non-linear partial differential equation (p.d.e.) constraints. It is assumed that the p.d.e. arises from minimizing a convex energy. We prove that the optimization problem is self-adjoint when the objective function is the dual energy. In other words, the differential of the objective function with respect to the optimization variable does not involve any adjoint state. This result generalizes the well-known fact that the so-called compliance is self-adjoint in the linear case. We also prove that in a large class of objective functions the dual energy is the only one which is self-adjoint.