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

Gao-Xi Li (), Xin-Min Yang () and Xian-Jun Long ()
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Gao-Xi Li: Chongqing Technology and Business University
Xin-Min Yang: Chongqing Normal University
Xian-Jun Long: Chongqing Technology and Business University

Journal of Global Optimization, 2025, vol. 92, issue 2, No 5, 345-379

Abstract: Abstract 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.

Keywords: Mathematical program with complementarity constraints; Non-Lipschitz continuity; Qualification; Augmented Lagrangian method; 90C26; 90C32; 90C33 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10898-024-01454-5

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