Avoiding the Use of Lagrange Multipliers: I—Evaluating the Constrained Extrema of Functions with Projection Matrices

David S. Corti () and Ricardo Fariello ()
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David S. Corti: Purdue University
Ricardo Fariello: State University of Montes Claros

SN Operations Research Forum, 2021, vol. 2, issue 4, 1-29

Abstract: Abstract A method for determining the extrema of a real-valued and differentiable function for which its dependent variables are subject to constraints is presented that avoids the use of Lagrange multipliers. The method makes use of projection matrices, and a corresponding Gram-Schmidt orthogonalization process, to identify the constrained extrema. Although Lagrange multipliers are not required, a comparison of our approach to the method of Lagrange multipliers nevertheless yields expressions for the Lagrange multipliers in terms of the gradients of the given function and the additional functions that represent the constraints. In addition, information about the second-derivatives of the given function with constraints is generated, from which the nature of the constrained extrema can be determined, again without knowledge of the Lagrange multipliers. Given its use of well-known aspects of linear algebra, the proposed method may prove computationally convenient for some optimization problems with a large number of constraints.

Keywords: Constrained extrema; Orthogonalization; Lagrange multipliers; Projection matrices; Determinants (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s43069-021-00100-0

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