Stochastic Programming - Index of

8 STOCHASTIC PROGRAMMING
(b) nonlinear, if at least one of the functions g i ,i=0, ···,m, is nonlinear or
X is not a convex polyhedral set; among nonlinear programs, we denote
a program as
(b1) convex, ifX ∩{x | g i (x) ≤ 0, i=1, ···,m} is a convex set and g 0 is
a convex function (in particular if the functions g i ,i=0, ···,m are
convex and X is a convex set); and
(b2) nonconvex, ifeitherX ∩{x | g i (x) ≤ 0, i=1, ···,m} is not a convex
set or the objective function g 0 is not convex.
Case (b2) above is also referred to as global optimization. Another special
class of problems, called (mixed) integer programs, arises if the set X requires
(at least some of) the variables x j , j =1, ···,n, to take integer values only.
We shall deal only briefly with discrete (i.e. mixed integer) problems, and
there is a natural interest in avoiding nonconvex programs whenever possible
for a very simple reason revealed by the following example from elementary
calculus.
Example 1.1 Consider the optimization problem
min ϕ(x), (2.4)
x∈IR
where ϕ(x) := 1 4 x4 − 5x 3 +27x 2 − 40x. A necessary condition for solving
problem (2.4) is
ϕ ′ (x) =x 3 − 15x 2 +54x − 40 = 0.
Observing that
ϕ ′ (x) =(x − 1)(x − 4)(x − 10),
we see that x 1 =1,x 2 =4andx 3 = 10 are candidates to solve our problem.
Moreover, evaluating the second derivative ϕ ′′ (x) =3x 2 − 30x + 54, we get
ϕ ′′ (x 1 )=27,
ϕ ′′ (x 2 )=−18,
ϕ ′′ (x 3 )=54,
indicating that x 1 and x 3 yield a relative minimum whereas in x 2 we find
a relative maximum. However, evaluating the two relative minima yields
ϕ(x 1 )=−17.75 and ϕ(x 3 )=−200. Hence, solving our little problem (2.4)
with a numerical procedure that intends to satisfy the first- and second-order
conditions for a minimum, we might (depending on the starting point of the
procedure) end up with x 1 as a “solution” without realizing that there exists
a (much) better possibility.
✷
As usual, a function ψ is said to attain a relative minimum—also called a
local minimum—at some point ˆx if there is a neighbourhood U of ˆx (e.g. a ball