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21.3.2 Conditions for Convexity
18.104.22.168 First-Order Condition
( ∂f (x )
, ∂f (x )
,..., ∂f (x )
dom f . Differentiable f is convex if and only
if dom f is convex and there holds
f(x) T (y
∀ x,y ∈
dom f . This is the first-order condition for convexity .
22.214.171.124 Second-Order Condition
2 f(x) (which
Function f is twice differentiable if dom f is open and the Hessian
∂ 2 f(x)
2 f(x) ij
is an n -by- n symmetric matrix with
∂x i ∂x j ,i,j =
1 , 2 ,...,n )exists
dom f . Twice differentiable f is convex if and only if dom f is convex and
2 f(x) is positive semidefinite (i.e.,
dom f .Thisisthe
second-order condition for convexity .
21.3.3 Convex Optimization Problem
An optimization problem is convex if the objective function is convex and the fea-
sible set is also convex. That is, a convex optimization problem is written as
g i (x)
h i (x)
where f and g i ,i
1 ,...,m are convex.
Note that any local optimum of a convex optimization problem is globally opti-
mal. Here are some examples of convex optimization problem.
Linear Programming (LP) A linear programming is a convex optimization prob-
lem with affine objective and constraint functions. Its feasible set is a polyhedron.
n α T x
g i (x)
h i ,i
x = b i ,i =