Mathematical methods for economic theory

Martin J. Osborne

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

For all the results, f and gj for j = 1, ..., m are continuously differentiable functions of n variables and cj 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 with one constraint

  ifg(x*) ≠ (0,...,0)  
x* solves
maxx f(x) subject to g(x) = c
there exists λ such that
L'i(x*) = 0 for i = 1, ..., n
and g(x*) = c
  if f is concave and λg is convex  

where L(x) = f(x) − λ(g(x) − c).

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 j = 1, ..., m
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=1
λj(gj(x) − cj).