A New Augmented Lagrangian Method for MPCCs—Theoretical and Numerical Comparison with Existing Augmented Lagrangian Methods

Lei Guo () and Zhibin Deng ()
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Lei Guo: School of Business, East China University of Science and Technology, Shanghai 200237, China
Zhibin Deng: School of Economics and Management, University of Chinese Academy of Sciences, Key Laboratory of Big Data Mining and Knowledge Management, Chinese Academy of Sciences, Beijing 100190, China

Mathematics of Operations Research, 2022, vol. 47, issue 2, 1229-1246

Abstract: We propose a new augmented Lagrangian (AL) method for solving the mathematical program with complementarity constraints (MPCC), where the complementarity constraints are left out of the AL function and treated directly. Two observations motivate us to propose this method: The AL subproblems are closer to the original problem in terms of the constraint structure; and the AL subproblems can be solved efficiently by a nonmonotone projected gradient method, in which we have closed-form solutions at each iteration. The former property helps us show that the proposed method converges globally to an M-stationary (better than C-stationary) point under MPCC relaxed constant positive linear dependence condition. Theoretical comparison with existing AL methods demonstrates that the proposed method is superior in terms of the quality of accumulation points and the strength of assumptions. Numerical comparison, based on problems in MacMPEC, validates the theoretical results.

Keywords: Primary: 90C33; 65K10; secondary: 90C59; mathematical program with complementarity constraints; augmented Lagrangian method; nonmonotone projected gradient method; M-stationarity (search for similar items in EconPapers)
Date: 2022
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