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<p>In recent years, mathematical programs with complementarity constraints (MPCC) and a non-Lipschitz objective function have been introduced and are now more prevalent than locally Lipschitz MPCC. This paper proposes a smoothing partial augmented Lagrangian (SPAL) method to tackle this problem. However, due to the disruption of the complementary structure’s integrity by this method, proving its convergence becomes exceptionally challenging. We have achieved global convergence of the SPAL method. Specifically, we demonstrate that the accumulation point of the sequence generated by the SPAL method can be a strongly stationary point under the Mangasarian-Fromovitz qualification (MPCC-MFQ) and the boundedness of the multiplier corresponding to the orthogonal constraint. Moreover, if the aforementioned multiplier is unbounded, the accumulation point can be a Clarke stationary point under MPCC-MFQ and a suitable assumption. Numerical experiments indicate that the SPAL method surpasses existing methods in terms of the quality of accumulation points and running times.</p>

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Partial augmented Lagrangian method for non-Lipschitz mathematical programs with complementarity constraints

  • 期刊论文

  • 发表日期：

    2025年1月3日

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  • Gao-Xi Li,
  • Xin-Min Yang,
  • Xian-Jun Long

摘要

In recent years, mathematical programs with complementarity constraints (MPCC) and a non-Lipschitz objective function have been introduced and are now more prevalent than locally Lipschitz MPCC. This paper proposes a smoothing partial augmented Lagrangian (SPAL) method to tackle this problem. However, due to the disruption of the complementary structure’s integrity by this method, proving its convergence becomes exceptionally challenging. We have achieved global convergence of the SPAL method. Specifically, we demonstrate that the accumulation point of the sequence generated by the SPAL method can be a strongly stationary point under the Mangasarian-Fromovitz qualification (MPCC-MFQ) and the boundedness of the multiplier corresponding to the orthogonal constraint. Moreover, if the aforementioned multiplier is unbounded, the accumulation point can be a Clarke stationary point under MPCC-MFQ and a suitable assumption. Numerical experiments indicate that the SPAL method surpasses existing methods in terms of the quality of accumulation points and running times.