Stochastic Programming - Index of

BASIC CONCEPTS 83
1.8.1 The Kuhn–Tucker Conditions
Remark 1.9 To get an idea of what kind of optimality conditions we may
expect for problems of the type (8.1), let us first—contrary to our general
assumption—consider the case where f,g i ,i=1, ···,m, are linear functions
}
f(x) :=c T x,
g i (x) :=a T i x − b (8.2)
i, i =1, ···,m,
such that we have the gradients
∇f(x) =c,
∇g i (x) =a i ,
and problem (8.1) becomes the linear program
min c T x
s.t. a T i x ≤ b i,
}
i =1, ···,m.
}
(8.3)
(8.4)
Although we did not explicitly discuss optimality conditions for linear
programs in the previous section, they are implicitly available in the duality
statements discussed there. The dual problem of (8.4) is
⎫
max{−b T u}
m∑
⎪⎬
s.t. − a i u i = c,
(8.5)
i=1
⎪⎭
u ≥ 0.
Let A be the m × n matrix having a T i , i =1, ···,m, as rows. The difference
of the primal and the dual objective functions can then be written as
c T x + b T u = c T x + u T Ax − u T Ax + b T u
=(c + A T u) T x +(b − Ax) T u
m∑
m∑
=[∇f(x)+ u i ∇g i (x)] T x − u i g i (x).
i=1
i=1
(8.6)
From the duality statements for linear programming (Propositions 1.17
and 1.18), we know the following.
(a) If ˆx is an optimal solution of the primal program (8.4) then, by the strong
duality theorem (Proposition 1.18), there exists a solution û of the dual
program (8.5) such that the difference of the primal and dual objective
vanishes. For the pair of dual problems (8.4) and (8.5) this means that
c Tˆx − (−b T û)=c Tˆx + b T û = 0. In view of (8.6) this may also be stated