# American Institute of Mathematical Sciences

January  2011, 7(1): 157-174. doi: 10.3934/jimo.2011.7.157

## Nonlinear augmented Lagrangian for nonconvex multiobjective optimization

 1 College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China, China 2 Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China 3 Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

Received  October 2009 Revised  October 2010 Published  January 2011

In this paper, based on the ordering relations induced by a pointed, closed and convex cone with a nonempty interior, we propose a nonlinear augmented Lagrangian dual scheme for a nonconvex multiobjective optimization problem by applying a class of vector-valued nonlinear augmented Lagrangian penalty functions. We establish the weak and strong duality results, necessary and sufficient conditions for uniformly exact penalization and exact penalization in the framework of nonlinear augmented Lagrangian. Our results include several ones in the literature as special cases.
Citation: Chunrong Chen, T. C. Edwin Cheng, Shengji Li, Xiaoqi Yang. Nonlinear augmented Lagrangian for nonconvex multiobjective optimization. Journal of Industrial & Management Optimization, 2011, 7 (1) : 157-174. doi: 10.3934/jimo.2011.7.157
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##### References:
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