A generalized Lagrangian function and multiplier method |
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Authors: | H Nakayama H Sayama Y Sawaragi |
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Institution: | (1) Department of Applied Mathematics, Faculty of Science, Konan University, Okamoto, Higashinadaku, Kobe, Japan;(2) Department of Industrial and Mechanical Engineering, School of Engineering, Okayama University, Okayama, Japan;(3) Department of Applied Mathematics and Physics, Faculty of Engineering, Kyoto University, Kyoto, Japan |
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Abstract: | As is well known, a saddle point for the Lagrangian function, if it exists, provides a solution to a convex programming problem; then, the values of the optimal primal and dual objective functions are equal. However, these results are not valid for nonconvex problems.In this paper, several results are presented on the theory of the generalized Lagrangian function, extended from the classical Lagrangian and the generalized duality program. Theoretical results for convex problems also hold for nonconvex problems by extension of the Lagrangian function. The concept of supporting hypersurfaces is useful to add a geometric interpretation to computational algorithms. This provides a basis to develop a new algorithm. |
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Keywords: | Augmented penalty function duality theory generalized Lagrangian function method of multipliers nonconvex programming |
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