Abstract
An exact augmented Lagrangian function for the nonlinear nonconvex programming problems with inequality constraints was discussed. Under suitable hypotheses, the relationship was established between the local unconstrained minimizers of the augmented Lagrangian function on the space of problem variables and the local minimizers of the original constrained problem. Furthermore, under some assumptions, the relationship was also established between the global solutions of the augmented Lagrangian function on some compact subset of the space of problem variables and the global solutions of the constrained problem. Therefore, from the theoretical point of view, a solution of the inequality constrained problem and the corresponding values of the Lagrange multipliers can be found by the well-known method of multipliers which resort to the unconstrained minimization of the augmented Lagrangian function presented.
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References
Bertsekas D P. Constrained Optimization and Lagrange Multipliers Methods[M]. Academic Press, New York, 1982.
Burke J. An exact penalization viewpoint of constrained optimization[J]. SIAM J Control Op-tim, 1991, 29(4):968–998.
Di Pillo G. Exact penalty methods [A]. In: Spedicato E (ed). Algorithms for Continuous Op-timization: the State of the Art[C]. Kluwer Academic Press, Boston, 1994, 209–253.
Di Pillo G, Grippo L. Exact penalty functions in constrained optimization [J]. SIAM J Control Optim, 1989,27(6):1333–1360.
Yevtushenko Y G, Zhadan V G. Exact auxiliary functions in optimization problems [J]. USSR Comput Maths and Math Phys, 1990, 30(1):31–42.
Contaldi G, Di Pillo G, Lucidi S. A continuously differentiable exact penalty function for non-linear programming problems with unbounded feasible set [J]. Oper Res Lett, 1993, 14(3): 153–161.
Di Pillo G, Grippo L. A continuously differentiable exact penalty function for nonlinear programming problems with inequality constraints[J]. SIAM J Control Optim, 1985, 23( 1):72–84.
Di Pillo G, Grippo L. On the exactness of a class of nondifferentiable penalty functions[J]. J Optim Theory Appl, 1988, 57(3):399–410.
Lucidi S. New results on a continuously differentiable exact penalty function[J]. SIAM J Optim, 1992, 2(4):558.
Di Pillo G, Grippo L. A new augmented Lagrangian function for inequality constraints in non-linear programming problems[J]. J Optim Theory Appl, 1982, 36(4):495–519.
Di Pillo G, Grippo L. A new class of augmented Lagrangians in nonlinear programming[J]. SIAM J Control Optim, 1979, 17(5):618–628.
Di Pillo G, Lucidi S. An augmented Lagrangian function with improved exactness properties [J]. SIAM J Optim, 2001, 12(2):376–406.
Di Pillo G, Lucidi S. On exact augmented Lagrangian functions in nonlinear programming [A]. In: Di Pillo G, Giannessi F (eds). Nonlinear Optimization and Applications [C]. Plenum Press, New York, 1996, 85–100.
Lucidi S. New results on a class of exact augmented Lagrangians [J]. J Optim Theory Appl, 1988, 58(2):259–282.
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Communicated by DAl Shi-qiang
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Du, Xw., Zhang, Ls., Shang, Yl. et al. Exact augmented lagrangian function for nonlinear programming problems with inequality constraints. Appl. Math. Mech.-Engl. Ed. 26, 1649–1656 (2005). https://doi.org/10.1007/BF03246275
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DOI: https://doi.org/10.1007/BF03246275
Key words
- local minimizer
- global minimizer
- nonlinear programming
- exact penalty function
- augmented Lagrangian function