# American Institute of Mathematical Sciences

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
##### References:
 [1] R. S. Burachik and A. Rubinov, Abstract convexity and augmented Lagrangians,, SIAM J. Optim., 18 (2007), 413. doi: 10.1137/050647621. [2] G. Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization: Set-Valued and Variational Analysis,", Springer, (2005). [3] F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley and Sons, (1983). [4] X. X. Huang and X. Q. Yang, A unified augmented Lagrangian approach to duality and exact penalization,, Math. Oper. Res., 28 (2003), 533. doi: 10.1287/moor.28.3.533.16395. [5] X. X. Huang and X. Q. Yang, Nonlinear Lagrangian for multiobjective optimization and applications to duality and exact penalization,, SIAM J. Optim., 13 (2002), 675. doi: 10.1137/S1052623401384850. [6] X. X. Huang and X. Q. Yang, Duality for multiobjective optimization via nonlinear Lagrangian functions,, J. Optim. Theory Appl., 120 (2004), 111. doi: 10.1023/B:JOTA.0000012735.86699.a1. [7] X. X. Huang and X. Q. Yang, Duality and exact penalization for vector optimization via augmented Lagrangian,, J. Optim. Theory Appl., 111 (2001), 615. doi: 10.1023/A:1012654128753. [8] X. X. Huang, X. Q. Yang and K. L. Teo, Convergence analysis of a class of penalty methods for vector optimization problems with cone constraints,, J. Global Optim., 36 (2006), 637. doi: 10.1007/s10898-004-1937-y. [9] J. Jahn, "Vector Optimization-Theory, Applications and Extensions,", Springer, (2004). [10] P. Q. Khanh, T. H. Nuong and M. Thera, On duality in nonconvex vector optimization in Banach spaces using augmented Lagrangians,, Positivity, 3 (1999), 49. doi: 10.1023/A:1009753224825. [11] A. Nedić and A. Ozdaglar, Separation of nonconvex sets with general augmenting functions,, Math. Oper. Res., 33 (2008), 587. doi: 10.1287/moor.1070.0296. [12] A. Nedić and A. Ozdaglar, A geometric framework for nonconvex optimization duality using augmented Lagrangian functions,, J. Global Optim., 40 (2008), 545. doi: 10.1007/s10898-006-9122-0. [13] R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis,", Springer-Verlag, (1998). doi: 10.1007/978-3-642-02431-3. [14] A. M. Rubinov, X. X. Huang and X. Q. Yang, The zero duality gap property and lower semicontinuity of the perturbation function,, Math. Oper. Res., 27 (2002), 775. doi: 10.1287/moor.27.4.775.295. [15] A. M. Rubinov and X. Q. Yang, "Lagrange-Type Functions in Constrained Non-Convex Optimization,", Kluwer Academic Publishers, (2003). [16] C. Singh, D. Bhatia and N. Rueda, Duality in nonlinear multiobjective programming using augmented Lagrangian functions,, J. Optim. Theory Appl., 88 (1996), 659. doi: 10.1007/BF02192203. [17] C. Y. Wang, X. Q. Yang and X. M. Yang, Zero duality gap and convergence of sub-optimal paths for optimization problems via a nonlinear augmented Lagrangian,, (2009) (preprint)., (2009). [18] C. Y. Wang, X. Q. Yang and X. M. Yang, Unified nonlinear Lagrangian approach to duality and optimal paths,, J. Optim. Theory Appl., 135 (2007), 85. doi: 10.1007/s10957-007-9225-x. [19] X. Q. Yang and X. X. Huang, A nonlinear Lagrangian approach to constrained optimization problems,, SIAM J. Optim., 11 (2001), 1119. doi: 10.1137/S1052623400371806. [20] Y. Y. Zhou and X. Q. Yang, Some results about duality and exact penalization,, J. Global Optim., 29 (2004), 497. doi: 10.1023/B:JOGO.0000047916.73871.88. [21] Y. Y. Zhou and X. Q. Yang, Augmented Lagrangian function, non-quadratic growth condition and exact penalization,, Oper. Res. Lett., 34 (2006), 127. doi: 10.1016/j.orl.2005.03.008. [22] Y. Y. Zhou and X. Q. Yang, Duality and penalization in optimization via an augmented Lagrangian function with applications,, J. Optim. Theory Appl., 140 (2009), 171. doi: 10.1007/s10957-008-9455-6.

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##### References:
 [1] R. S. Burachik and A. Rubinov, Abstract convexity and augmented Lagrangians,, SIAM J. Optim., 18 (2007), 413. doi: 10.1137/050647621. [2] G. Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization: Set-Valued and Variational Analysis,", Springer, (2005). [3] F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley and Sons, (1983). [4] X. X. Huang and X. Q. Yang, A unified augmented Lagrangian approach to duality and exact penalization,, Math. Oper. Res., 28 (2003), 533. doi: 10.1287/moor.28.3.533.16395. [5] X. X. Huang and X. Q. Yang, Nonlinear Lagrangian for multiobjective optimization and applications to duality and exact penalization,, SIAM J. Optim., 13 (2002), 675. doi: 10.1137/S1052623401384850. [6] X. X. Huang and X. Q. Yang, Duality for multiobjective optimization via nonlinear Lagrangian functions,, J. Optim. Theory Appl., 120 (2004), 111. doi: 10.1023/B:JOTA.0000012735.86699.a1. [7] X. X. Huang and X. Q. Yang, Duality and exact penalization for vector optimization via augmented Lagrangian,, J. Optim. Theory Appl., 111 (2001), 615. doi: 10.1023/A:1012654128753. [8] X. X. Huang, X. Q. Yang and K. L. Teo, Convergence analysis of a class of penalty methods for vector optimization problems with cone constraints,, J. Global Optim., 36 (2006), 637. doi: 10.1007/s10898-004-1937-y. [9] J. Jahn, "Vector Optimization-Theory, Applications and Extensions,", Springer, (2004). [10] P. Q. Khanh, T. H. Nuong and M. Thera, On duality in nonconvex vector optimization in Banach spaces using augmented Lagrangians,, Positivity, 3 (1999), 49. doi: 10.1023/A:1009753224825. [11] A. Nedić and A. Ozdaglar, Separation of nonconvex sets with general augmenting functions,, Math. Oper. Res., 33 (2008), 587. doi: 10.1287/moor.1070.0296. [12] A. Nedić and A. Ozdaglar, A geometric framework for nonconvex optimization duality using augmented Lagrangian functions,, J. Global Optim., 40 (2008), 545. doi: 10.1007/s10898-006-9122-0. [13] R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis,", Springer-Verlag, (1998). doi: 10.1007/978-3-642-02431-3. [14] A. M. Rubinov, X. X. Huang and X. Q. Yang, The zero duality gap property and lower semicontinuity of the perturbation function,, Math. Oper. Res., 27 (2002), 775. doi: 10.1287/moor.27.4.775.295. [15] A. M. Rubinov and X. Q. Yang, "Lagrange-Type Functions in Constrained Non-Convex Optimization,", Kluwer Academic Publishers, (2003). [16] C. Singh, D. Bhatia and N. Rueda, Duality in nonlinear multiobjective programming using augmented Lagrangian functions,, J. Optim. Theory Appl., 88 (1996), 659. doi: 10.1007/BF02192203. [17] C. Y. Wang, X. Q. Yang and X. M. Yang, Zero duality gap and convergence of sub-optimal paths for optimization problems via a nonlinear augmented Lagrangian,, (2009) (preprint)., (2009). [18] C. Y. Wang, X. Q. Yang and X. M. Yang, Unified nonlinear Lagrangian approach to duality and optimal paths,, J. Optim. Theory Appl., 135 (2007), 85. doi: 10.1007/s10957-007-9225-x. [19] X. Q. Yang and X. X. Huang, A nonlinear Lagrangian approach to constrained optimization problems,, SIAM J. Optim., 11 (2001), 1119. doi: 10.1137/S1052623400371806. [20] Y. Y. Zhou and X. Q. Yang, Some results about duality and exact penalization,, J. Global Optim., 29 (2004), 497. doi: 10.1023/B:JOGO.0000047916.73871.88. [21] Y. Y. Zhou and X. Q. Yang, Augmented Lagrangian function, non-quadratic growth condition and exact penalization,, Oper. Res. Lett., 34 (2006), 127. doi: 10.1016/j.orl.2005.03.008. [22] Y. Y. Zhou and X. Q. Yang, Duality and penalization in optimization via an augmented Lagrangian function with applications,, J. Optim. Theory Appl., 140 (2009), 171. doi: 10.1007/s10957-008-9455-6.
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