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<p>Recently, preconditioned primal-dual methods with projection (PPP) have gained popularity for solving inclusion problems. However, existing algorithms often lack concrete guidelines for selecting general nonlinear preconditioners and provide limited numerical validation. To address these issues, we propose a nonlinear preconditioned primal-dual with projection (NL-PPP) method. This framework offers explicit strategies for designing strongly monotone, nonlinear, and potentially nonsymmetric preconditioners; it is applicable to a broader class of nonmonotone inclusion problems characterized by the weak MVI (Minty Variational Inequality) condition. Moreover, NL-PPP not only encompasses existing preconditioners but also integrates previous PPP algorithms with nonlinear preconditioners into a unified framework. The convergence analysis of NL-PPP is established by leveraging the projective correction underlying the separate and project principle. Numerical experiments on several minimax problems demonstrate the effectiveness and flexibility of the proposed NL-PPP.</p>

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Nonlinear Preconditioned Primal-Dual Method with Projection for Nonconvex-Nonconcave Minimax Problems

  • 期刊论文

  • 发表日期：

    2026年5月7日

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  • Lu Zhang,
  • Feng Xue,
  • Hongxia Wang,
  • Hui Zhang

摘要

Recently, preconditioned primal-dual methods with projection (PPP) have gained popularity for solving inclusion problems. However, existing algorithms often lack concrete guidelines for selecting general nonlinear preconditioners and provide limited numerical validation. To address these issues, we propose a nonlinear preconditioned primal-dual with projection (NL-PPP) method. This framework offers explicit strategies for designing strongly monotone, nonlinear, and potentially nonsymmetric preconditioners; it is applicable to a broader class of nonmonotone inclusion problems characterized by the weak MVI (Minty Variational Inequality) condition. Moreover, NL-PPP not only encompasses existing preconditioners but also integrates previous PPP algorithms with nonlinear preconditioners into a unified framework. The convergence analysis of NL-PPP is established by leveraging the projective correction underlying the separate and project principle. Numerical experiments on several minimax problems demonstrate the effectiveness and flexibility of the proposed NL-PPP.