Convergence of the Implicit Filtering Method for Constrained Optimization of Noisy Functions

Abstract

In this paper, we develop convergence theory of the implicit filtering method for solving the box constrained optimization whose objective function includes a smooth term and a noisy term. It is shown that under certain assumption on the noisy function, the sequence of projected gradients on the smooth function produced by the method goes to zero. Moreover, it is shown that if the smooth function is convex and the noisy function decays near optimality, the whole sequence of iterates converges to a solution of the concerned problem and possesses the finite identification for the optimal active set under the nondegenerate assumption. Finally, preliminary numerical results are reported.

Keywords:

• Convergence theory
• Implicit filtering
• Noisy optimization
• Projection

AMS Subject Classification:

• 90C30
• 90C33
• 65K05

ACKNOWLEDGMENTS

The authors would like to thank the referees for their useful comments and suggestions.

This research was partly supported by the National Natural Science Foundation of China (70471002, NCET040098), and the CityU Strategic Research Grant (7001427).