Abstract
The proximal method of multipliers was proposed by Rockafellar (Math Oper Res 1:97–116, 1976) for solving convex programming and it is a kind of proximal point method for convex programming. In this paper, we apply this method for equality constrained optimization problems, in which subproblems have better properties than those from the augmented Lagrange method. We prove that, under linear independence constraint qualification and the second-order sufficiency optimality condition, the rate of convergence of the proximal method of multipliers, for the equality constrained optimization problem, is linear and the ratio constant is proportional to 1 / c, where c is the penalty parameter that exceeds a threshold \(c_*> 0\). Moreover, the rate of convergence of the proximal method of multipliers is superlinear when the parameter c increases to \(+\infty \).
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Acknowledgements
The authors are grateful to the Associate Editor and the referees for their helpful comments and suggestions on improving the quality of this paper. This work was supported by the National Natural Science Foundation of China under Project Nos. 11571059 and 11731013.
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Zhang, Y., Wu, J. & Zhang, L. The rate of convergence of proximal method of multipliers for equality constrained optimization problems. Optim Lett 14, 1599–1613 (2020). https://doi.org/10.1007/s11590-019-01449-2
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DOI: https://doi.org/10.1007/s11590-019-01449-2