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| Volume 2, Number 3, September 2006, pp. 501-519 | |||||||||
| Joydeep Dutta | |||||||||
| Key words: | |||||||||
| Lagrange multipliers, nonsmooth optimization, subdifferential, locally Lipschitz functions, penalty method, normal cone | |||||||||
| Mathematices Subject Classification: 0C30, 90C46, 49J52 | |||||||||
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| Volume 2, Number 3, September 2006, pp. 501-519 | |||||||||
| Joydeep Dutta | |||||||||
| Key words: | |||||||||
| Lagrange multipliers, nonsmooth optimization, subdifferential, locally Lipschitz functions, penalty method, normal cone | |||||||||
| Mathematices Subject Classification: 0C30, 90C46, 49J52 | |||||||||
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| Abstract: | |||
| In this article our main aim is to study the Lagrange multiplier rule associated with a very general class of optimization problem which goes much beyond the standard framework using only equality and inequality constraints. Traditionally the algorithms are developed by taking inputs from the theory. However the penalty method which is heavily used in practice has also been applied to derive the Lagrange multiplier rule associated with equality and inequality problem. In this article we use a penalty approximation approach due to Rockafellar and apply it to derive the Lagrange multiplier rule for a general class of optimization problems. This approach is interesting since it is inherently simple and at the same time one can figure out from the proof the qualification condition required for the problem. The other approaches to the proof of this problem is more involved and requires much more technical sophistication. On our way to the main result we will give a detailed motivation as to why such an approach is taken and its pedegogical value. We will also provide a free standing exposition to nonsmooth analysis and nonsmooth geometry that is required to derive the Lagrange multiplier rule for the problem under consideration. |
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| Revisiting the lagrange multiplier rule | ||