Stochastic Programming - Index of

BASIC CONCEPTS 89
and observing that ∇ u L(x, u) ≤ 0 simply repeats the constraints g i (x) ≤ 0 ∀i
of our original program (8.1), the Kuhn–Tucker conditions now read as
⎫
∇ x L(ˆx, û) =0, ⎪⎬
∇ u L(ˆx, û) ≤ 0,
û T (8.10)
∇ u L(ˆx, û) =0, ⎪⎭
û ≥ 0.
Assume now that the functions f,g i , i =1, ···,m, are convex. Then for
any fixed u ≥ 0 the Lagrange function is obviously convex in x. For(ˆx, û)
satisfying the Kuhn–Tucker conditions, it follows by Proposition 1.21 that for
any arbitrary x
L(x, û) − L(ˆx, û) ≥ (x − ˆx) T ∇ x L(ˆx, û) =0
and hence
L(ˆx, û) ≤ L(x, û) ∀x ∈ IR n .
On the other hand, since ∇ u L(ˆx, û) ≤ 0isequivalenttog i (ˆx) ≤ 0 ∀i, and
the Kuhn–Tucker conditions assert that û T ∇ u L(ˆx, û) = ∑ m
i=1 ûig i (ˆx) =0,it
follows that
L(ˆx, u) ≤ L(ˆx, û) ∀u ≥ 0.
Hence we have the following.
Proposition 1.25 Given that the functions f,g i , i =1, ···,m, in problem
(8.1) are convex, any Kuhn–Tucker point, i.e. any pair (ˆx, û) satisfying the
Kuhn–Tucker conditions, is a saddle point of the Lagrange function, i.e. it
satisfies
∀u ≥ 0 L(ˆx, u) ≤ L(ˆx, û) ≤ L(x, û) ∀x ∈ IR n .
Furthermore, it follows by the complementarity conditions that
L(ˆx, û) =f(ˆx).
It is an easy exercise to show that for any saddle point (ˆx, û), with û ≥ 0,
of the Lagrange function, the Kuhn–Tucker conditions (8.10) are satisfied.
Therefore, if we knew the right multiplier vector û in advance, the task to
solve the constrained optimization problem (8.1) would be equivalent to
that of solving the unconstrained optimization problem min x∈IR n L(x, û). This
observation can be seen as the basic motivation for the development of a class
of solution techniques known in the literature as Lagrangian methods.
1.8.2 Solution Techniques
When solving stochastic programs, we need to use known procedures from
both linear and nonlinear programming, or at least adopt their underlying