Abstract
This paper establishes two sets of "second order" conditions-one which is necessary, the other which is sufficient-in order that a vector x* be a local minimum to the constrained optimization problem: minimize f(x) subject to the constraints \( g_{i}(x)\geqq 0,i=1,\cdots ,m,\; \rm{and} \; h_{i}(x)=0,j=1,\cdots,p, \) where the problem functions are twice continuously differentiable. The necessary conditions extend the well-known results, obtained with Lagrange multipliers, which apply to equality constrained optimization problems, and the Kuhn-Tucker conditions, which apply to mixed inequality and equality problems when the problem functions are required only to have continuous first derivatives. The sufficient conditions extend similar conditions which have been developed only for equality constrained problems. Examples of the applications of these sets of conditions are given.
Received by the editors April 29, 1966.
Advanced Research Department, Research Analysis Corporation, McLean, Virginia 22101.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Basel
About this chapter
Cite this chapter
McCormick, G.P. (2014). Second Order Conditions for Constrained Minima. In: Giorgi, G., Kjeldsen, T. (eds) Traces and Emergence of Nonlinear Programming. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0439-4_12
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0439-4_12
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0438-7
Online ISBN: 978-3-0348-0439-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)