7.5 Optimization: summary of conditions under which firstorder conditions are necessary and sufficient
Unconstrained maximization problems
x* solves max_{x} f(x)  f'_{i}(x*) = 0 for i = 1, ..., n  
if f is concave 
Equalityconstrained maximization problems with one constraint
if ∇g(x*) ≠ (0,...,0)  
x* solves max_{x} 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).
Inequalityconstrained maximization problems
if g_{j} is concave for j = 1, ..., m or g_{j} is convex for j = 1, ..., m and there exists x such that g_{j}(x) < c_{j} for j = 1, ..., m or g_{j} is quasiconvex for j = 1, ..., m, ∇g_{j}(x*) ≠ (0,...,0) for j = 1, ..., m, and there exists x such that g_{j}(x) < c_{j} for j = 1, ..., m 

x* solves max_{x} f(x) subject to g_{j}(x) ≤ c_{j} for j = 1, ..., m 
there exists (λ_{1},...,λ_{m}) such that L'_{i}(x*) = 0 for i = 1, ..., n and λ_{j} ≥ 0, g_{j}(x*) ≤ c_{j}, and λ_{j}(g_{j}(x*) − c_{j}) = 0 for j = 1, ..., m 

if g_{j} 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}).