On second order conditions for equality constrained extremum problems

We prove a relationship between the bordered Hessian in an equality constrained extremum problem and the Hessian of the equivalent lower-dimension unconstrained problem. This relationship can be used to derive principal minor conditions for the former from the relatively simple and accessible condit...

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Bibliographic Details
Published in:Economics letters Vol. 121; no. 3; pp. 440 - 443
Main Author: Mandy, David M.
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 01.12.2013
Elsevier Science Ltd
Subjects:
Economic analysis
Economic conditions
Economic theory
economics
Equality
Hessian
Mathematical analysis
Mathematical problems
Matrix
Minors
Optimization
Studies
C60
Hessian
Optimization
Minors
ISSN:0165-1765, 1873-7374
Online Access:Get full text
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Summary:We prove a relationship between the bordered Hessian in an equality constrained extremum problem and the Hessian of the equivalent lower-dimension unconstrained problem. This relationship can be used to derive principal minor conditions for the former from the relatively simple and accessible conditions for the latter. •Establishes a simple relationship between a Hessian and bordered Hessian.•Derives necessary and sufficient second order conditions from this relationship.•The only proof that avoids use of quadratic forms subject to side conditions.•Clarifies Samuelson’s “great loss of symmetry”.
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ISSN:0165-1765
1873-7374
DOI:10.1016/j.econlet.2013.09.017