Centre for Discrete and Applicable Mathematics
	CDAM Research Report, LSE-CDAM-2003-15 December 2003

Abstract

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.

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