The rate of convergence of proximal method of multipliers for nonlinear semidefinite programming

ABSTRACT

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 $c_{\ast }>0$. Moreover, the rate of convergence of the proximal method of multipliers is superlinear when the parameter c increases to $+\mathrm{\infty }$.

KEYWORDS:

• Nonlinear semidefinite programming
• rate of convergence
• the proximal method of multipliers

AMS SUBJECT CLASSIFICATION:

• 90C30

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China under projects Nos. 11571059 and 11731013.