Abstract
In the first part of this paper series, a unified duality scheme for a constrained extremum problem is proposed by virtue of the image space analysis. In the present paper, we pay our attention to study of some special duality schemes. Particularly, the Lagrange-type duality, Wolfe duality and Mond–Weir duality are discussed as special duality schemes in a unified interpretation. Moreover, three practical classes of regular weak separation functions are also considered.
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Castellani, G., Giannessi, F.: Decomposition of mathematical programs by means of theorems of alternative for linear and nonlinear systems. In: Proc. Ninth Internat. Math. Programming Sympos., Budapest. Survey of Mathematical Programming, pp. 423–439. North-Holland, Amsterdam (1979)
Giannessi, F.: Theorems of the alternative and optimality conditions. J. Optim. Theory Appl. 42, 331–365 (1984)
Zhu, S.K., Li, S.J.: Unified duality theory for constrained extremum problems. Part I: Image space analysis approach. J. Optim. Theory Appl. (2013). doi:10.1007/s10957-013-0468-4
Hestenes, M.R.: Multiplier and gradient methods. In: Zadeh, L.A., Neustadt, L.W., Balakrishnan, A.V. (eds.) Computing Methods in Optimization Problems. Academic Press, New York (1969)
Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969)
Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization. Academic Press, New York (1972)
Rubinov, A.M., Glover, B.M., Yang, X.Q.: Decreasing functions with applications to penalization. SIAM J. Optim. 10, 289–313 (1999)
Huang, X.X., Yang, X.Q.: A unified augmented Lagrangian approach to duality and exact penalization. Math. Oper. Res. 28, 533–552 (2003)
Wang, C.Y., Yang, X.Q., Yang, X.M.: Nonlinear Lagrange duality theorems and penalty function methods in continuous optimization. J. Glob. Optim. 27, 473–484 (2003)
Giannessi, F., Mastroeni, G.: Separation of sets and Wolfe duality. J. Glob. Optim. 42, 401–412 (2008)
Giannessi, F.: Constrained Optimization and Image Space Analysis, vol. 1: Separation of Sets and Optimality Conditions. Springer, New York (2005)
Giannessi, F.: On the theory of Lagrangian duality. Optim. Lett. 1, 9–20 (2007)
Mastroeni, G.: Some applications of the image space analysis to the duality theory for constrained extremum problems. J. Glob. Optim. 46, 603–614 (2010)
Luo, H.Z., Wu, H.X., Liu, J.Z.: Some results on augmented Lagrangians in constrained global optimization via image space analysis. J. Optim. Theory Appl. 159, 360–385 (2013)
Li, J., Feng, S.Q., Zhang, Z.: A unified approach for constrained extremum problems: image space analysis. J. Optim. Theory Appl. 159, 69–92 (2013)
Pappalardo, M.: Image space approach to penalty methods. J. Optim. Theory Appl. 64, 141–152 (1990)
Mastroeni, G.: Nonlinear separation in the image space with applications to penalty methods. Appl. Anal. 91, 1901–1914 (2012)
Rubinov, A.M., Uderzo, A.: On global optimality conditions via separation functions. J. Optim. Theory Appl. 109, 345–370 (2001)
Rubinov, A.M., Yang, X.Q.: Lagrange-Type Functions in Constrained Non-convex Optimization. Kluwer Academic, Dordrecht (2003)
Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)
Wolfe, P.: A duality theorem for nonlinear programming. Q. Appl. Math. 19, 239–244 (1961)
Mond, B., Weir, T.: Generalized concavity and duality. In: Generalized Concavity in Optimization, pp. 263–279. Academic Press, New York (1981)
Luo, H.Z., Mastroeni, G., Wu, H.X.: Separation approach for augmented Lagrangians in constrained nonconvex optimization. J. Optim. Theory Appl. 144, 275–290 (2010)
Yang, X.Q., Huang, X.X.: A nonlinear Lagrangian approach to constrained optimization problems. SIAM J. Optim. 14, 1119–1144 (2001)
Rubinov, A.M., Huang, X.X., Yang, X.Q.: The zero duality gap property and lower semicontinuity of the perturbation function. Math. Oper. Res. 27, 775–791 (2002)
Wang, C.Y., Yang, X.Q., Yang, X.M.: Unified nonlinear Lagrangian approach to duality and optimal paths. J. Optim. Theory Appl. 135, 85–100 (2007)
Sun, X.L., Li, D., Mckinnon, K.I.M.: On saddle points of augmented Lagrangians for constrained nonconvex optimization. SIAM J. Optim. 15, 1128–1146 (2005)
Li, S.J., Xu, Y.D., Zhu, S.K.: Nonlinear separation approach to constrained extremum problems. J. Optim. Theory Appl. 154, 842–856 (2012)
Hiriart-Urruty, J.B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4, 79–97 (1979)
Hiriart-Urruty, J.B.: New concepts in nondifferentiable programming, analysenonconvexe. Bull. Soc. Math. Fr. 60, 57–85 (1979)
Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42, 1071–1086 (2003)
Durea, M., Dutta, J., Tammer, C.: Lagrange multipliers for ε-Pareto solutions in vector optimization with non solid cones in Banach spaces. J. Optim. Theory Appl. 145, 196–211 (2010)
Durea, M., Strugariu, R.: Necessary optimality conditions for weak sharp minima in set-valued optimization. Nonlinear Anal. 73, 2148–2157 (2010)
Gasimov, R.N., Rubinov, A.M.: On augmented Lagrangians for optimization problems with a single constraint. J. Glob. Optim. 28, 153–173 (2004)
Acknowledgements
The authors are grateful to the two anonymous referees and Professor F. Giannessi for their valuable comments and suggestions, especially for providing the references [12, 14, 15], which helped to improve the paper. This research was supported by the National Natural Science Foundation of China (Grant: 11171362) and the Basic and Advanced Research Project of CQCSTC (Grant: cstc2013jcyjA00003).
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Zhu, S.K., Li, S.J. Unified Duality Theory for Constrained Extremum Problems. Part II: Special Duality Schemes. J Optim Theory Appl 161, 763���782 (2014). https://doi.org/10.1007/s10957-013-0467-5
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DOI: https://doi.org/10.1007/s10957-013-0467-5