Nonlinear rescaling Lagrangians for nonconvex semidefinite programming

Abstract

This paper focuses on the study of rescaling Lagrangians for solving nonconvex semidefinite programming problems. The rescaling nonlinear Lagrangians are generated by Löwner operators associated with convex real-valued functions. A set of conditions on the convex real-valued functions is proposed to guarantee the convergence of nonlinear rescaling Lagrangian algorithms. These conditions are satisfied by well-known nonlinear Lagrangians appeared in the literature. The convergence theorem shows that, under the second-order sufficient conditions with sigma-term and the strict constraint nondegeneracy condition, the nonlinear rescaling Lagrange algorithm is locally convergent when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter. Compared to the analysis in the nonlinear rescaling Lagrangian method for nonlinear programming, we have to deal with the sigma term in the convergence analysis.

Keywords:

• semidefinite programming
• nonlinear Lagrangian
• augmented Lagrangian
• dual algorithm

AMS Subject Classifications:

• 90C22
• 65K05

Acknowledgments

The authors thank the referees for their careful readings of the paper and helpful suggestions.

Notes

Dedicated to Professor Yue Minyi on the Occasion of his 95th Birthday.

1 One may consult [26, Proposition 5.71] for a proof, where Bonnans and Shapiro only considered the case that . However, it is easy to modify their arguments to include the equality constraint .

Supported by the National Natural Science Foundation of China [project numbers 11071029, 91130007, 91330206 and 11301049], Tian Yuan Natural Science Foundation of China [project number 11226223] and the Fundamental Research Funds for the Central Universities [projects numbers DC120101113 and DUT12RC(3)51].