Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2003-15

December 2003

First and second-order conditions in constrained optimisation

Anthony Horsley and Andrew J. Wrobel


We give a concise but complete and detailed exposition of the "classical" approach, based on directional variations, to the first-order and second-order conditions (FOC and SOCs) for finite-dimensional constrained optimisation with both equality and inequality constraints. Attention is paid to liminal constraints, which are active inequality constraints with zero Lagrange multipliers. The persistent assertion in economics texts that all active constraints can be treated like equality constraints is untrue with liminal constraints, and it gives a false "sufficient" SOC. Nor can liminal constraints be ignored like inactive constraints; this would give a false "necessary" SOC. Treating liminal constraints like equalities in the necessary SOC, or ignoring them in the sufficient SOC is not incorrect, but it gives weaker optimality criteria than the standard Kuhn-Tucker multiplier rules (although the resulting Strong Sufficient SOC does have a place in the directional-derivative results of solution-sensitivity analysis without strict complementarity). We also show that the square slack-variables method, which reduces inequalities to equalities, cannot deal properly with liminal constraints in SOCs.

A PDF file (473 kB) with the full contents of this report can be downloaded by clicking here.

Alternatively, if you would like to get a free hard copy of this report, please send the number of this report, LSE-CDAM-2003-15, together with your name and postal address to: Sorry, this report is currently yet not available in electronic format. To obtain a free hard copy, please send the number of this report, LSE-CDAM-2003-15, together with your name and postal address to:
CDAM Research Reports Series
Centre for Discrete and Applicable Mathematics
London School of Economics
Houghton Street
London WC2A 2AE, U.K.
Phone: +44(0)-20-7955 7732.
Fax: +44(0)-20-7955 6877.
Email: info@maths.lse.ac.uk

Introduction to the CDAM Research Report Series.
CDAM Homepage.

Copyright © London School of Economics & Political Science 2005

Last changed: Wed 9 Feb 2005
For comments go to: http://www.maths.lse.ac.uk/webmaster.html