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We analyze the rate of convergence of the proximal method of multipliers for non-convex nonlinear programming problems. First, we prove, under the strict complementarity condition, that the rate of convergence of the proximal method of multipliers is linear and the ratio constant is proportional to 1/c when the ratio ‖ (μ 0 , λ 0) − (μ ¯ , λ ¯) ‖ / c is small enough, which implies that the rate of convergence of the proximal method of multipliers is superlinear when the parameter c increases to + ∞. Second, we prove that, without strict complementarity condition, the rate of convergence of the proximal method of multipliers is proportional to 1/c when c exceeds a threshold. [ABSTRACT FROM AUTHOR]

The proximal method of multipliers was proposed by Rockafellar [Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math Oper Res. 1976;1:97–116] for solving convex programming and it is a kind of proximal point method applied to convex programming. In this paper, we apply this method for solving nonlinear semidefinite programming problems, in which subproblems have better properties than those from the augmented Lagrange method. We prove that, under the linear independence constraint qualification and the strong second-order sufficiency optimality condition, the rate of convergence of the proximal method of multipliers, for a nonlinear semidefinite programming 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 + ∞. [ABSTRACT FROM AUTHOR]