American Institute of Mathematical Sciences

Advanced
2011, 1(3): 539-548. doi: 10.3934/naco.2011.1.539

Asymptotic strong duality

Regina S. Burachik 1, and  Xiaoqi Yang 2,

1. 

School of Mathematics and Statistics, University of South Australia, SA 5095, Australia
2. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

Received  June 2011 Revised  August 2011 Published  September 2011

Given a nonconvex and nonsmooth optimization problem, we define a family of ``perturbed'' Lagrangians, which induce well-behaved approximations of the dual problem. Our family of approximated problems is said to verify {\em strong asymptotic duality} when the optimal dual values of the perturbed problems approach the primal optimal value. Our perturbed Lagrangians can have the same order of smoothness as the functions of the original problem, a property not shared by the classical (unperturbed) augmented Lagrangian. Therefore our proposed scheme allows the use of efficient numerical methods for solving the perturbed dual problems. We establish general conditions under which strong asymptotic duality holds, and we relate the latter with both strong duality and lower semicontinuity of the perturbation function. We illustrate our perturbed duality scheme with two important examples: Constrained Nonsmooth Optimization and Nonlinear Semidefinite programming.
Keywords: duality schemes, nonsmooth optimization, augmented Lagrangians, Nonconvex programming, nonlinear semidefinite programming..
Mathematics Subject Classification: Primary: 90C26, 49M29, 49M37, 90C9.
Citation: Regina S. Burachik, Xiaoqi Yang. Asymptotic strong duality. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 539-548. doi: 10.3934/naco.2011.1.539
References:
[1]

K. M. Abadir and J. R. Magnus, "Matrix Algebra,", Cambridge University Press, (2005). Google Scholar

[2]

X. X. Huang, K. L. Teo and X. Q. Yang, Approximate Augmented Lagrangian Functions and Nonlinear Semidefinite Programs,, Acta Mathematica Sinica, 22 (2006), 1283. doi: 10.1007/s10114-005-0702-6. Google Scholar

[3]

R. T. Rockafellar and R. J. B. Wets, "Variational Analysis,", Springer, (1998). doi: 10.1007/978-3-642-02431-3. Google Scholar

[4]

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. Google Scholar

[5]

C. Y. Wang, X. Q. Yang and X. M. Yang, Unified nonlinear Lagrangian approach to duality and optimal paths,, J. Optimiz. Theory Appl., 135 (2007), 85. doi: 10.1007/s10957-007-9225-x. Google Scholar

show all references

References:
[1]

K. M. Abadir and J. R. Magnus, "Matrix Algebra,", Cambridge University Press, (2005). Google Scholar

[2]

X. X. Huang, K. L. Teo and X. Q. Yang, Approximate Augmented Lagrangian Functions and Nonlinear Semidefinite Programs,, Acta Mathematica Sinica, 22 (2006), 1283. doi: 10.1007/s10114-005-0702-6. Google Scholar

[3]

R. T. Rockafellar and R. J. B. Wets, "Variational Analysis,", Springer, (1998). doi: 10.1007/978-3-642-02431-3. Google Scholar

[4]

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. Google Scholar

[5]

C. Y. Wang, X. Q. Yang and X. M. Yang, Unified nonlinear Lagrangian approach to duality and optimal paths,, J. Optimiz. Theory Appl., 135 (2007), 85. doi: 10.1007/s10957-007-9225-x. Google Scholar

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