Mathematical methods for economic theory

7.5 Optimization: summary of conditions under which first-order conditions are necessary and sufficient

For all the results, f and g_j for j = 1, ..., m are continuously differentiable functions of n variables defined on an open set and c_j for j = 1, ..., m are constants.

Unconstrained maximization problems

x* solves maxx f(x)		f'i(x*) = 0 for i = 1, ..., n
	if f is concave

Equality-constrained maximization problems

	if rank of Jacobian of g1, ..., gm is m
x* solves maxx f(x) subject to gj(x) = cj for j = 1, ..., m		there exist λ1, ..., λm such that L'i(x*) = 0 for i = 1, ..., n and gj(x*) = cj for j = 1, ..., m
	if f is concave and λg is convex

where L(x) = f(x) − ∑m
j=1λ_j(g_j(x) − c_j).

Inequality-constrained maximization problems

if gj is concave for j = 1, ..., m or gj is convex for j = 1, ..., m and there exists x such that gj(x) ≤ cj for every j for which gj is linear and gj(x) < cj for every j for which gj is not linear or gj is quasiconvex for j = 1, ..., m, ∇gj(x*) ≠ (0,...,0) for j = 1, ..., m, and there exists x such that gj(x) < cj for j = 1, ..., m
x* solves maxx f(x) subject to gj(x) ≤ cj for j = 1, ..., m		there exists (λ1,...,λm) such that L'i(x*) = 0 for i = 1, ..., n and λj ≥ 0, gj(x*) ≤ cj, and λj(gj(x*) − cj) = 0 for j = 1, ..., m
if gj is quasiconvex for j = 1, ..., m and either f is concave or f is quasiconcave and ∇f(x*) ≠ (0,...,0)

where L(x) = f(x) − ∑m
j=1λ_j(g_j(x) − c_j).