Stochastic Programming - Index of

98 STOCHASTIC PROGRAMMING
(presumably unique) solution ˆx of problem (8.1) some constraint i 0 ∈ I is
active, i.e. g i0 (ˆx) =0.Foranyfixedr>0, minimization of (8.13) will not
allow us to approach the solution ˆx, since obviously, by the definition of a
barrier function, this would drive the new objective F rs to +∞. Hence it
seems reasonable to drive the parameter r downwards to zero, as well.
With
B 1 := {x | g i (x) ≤ 0, i∈ I}, B 2 := {x | g i (x) ≤ 0, i∈ J}
we have B = B 1 ∩B 2 ,andforr > 0 we may expect finite values of F rs
only for x ∈ B1
0 := {x | g i(x) < 0, i ∈ I}. We may close this short
presentation of general penalty methods by a statement showing that, under
mild assumptions, a method of this type may be controlled in such a way that
it results in what we should like to experience.
Proposition 1.26 Let f,g i , i =1, ···,m, be convex and assume that
B 0 1 ∩B 2 ≠ ∅
and that B = B 1 ∩B 2 is bounded. Then for {r k } and {s k } strictly monotone
sequences decreasing to zero there exists an index k 0 such that for all k ≥ k 0
the modified objective function F rk s k
attains its (free) minimum at some point
x (k) where x (k) ∈B1.
0
The sequence {x (k) | k ≥ k 0 } is bounded, and any of its accumulation points
is a solution of the original problem (8.1). With γ the optimal value of (8.1),
the following relations hold:
lim
k→∞ f(x(k) ) = γ,
lim r ∑
k ϕ(g i (x (k) )) = 0,
k→∞
i∈I
1 ∑
lim ψ(g i (x (k) )) = 0.
k→∞ s k
i∈J
1.8.2.4 Lagrangian methods
As mentioned at the end of Section 1.8.1, knowledge of the proper multiplier
vector û in the Lagrange function L(x, u) = f(x) + ∑ m
i=1 u ig i (x) for
problem (8.1) would allow us to solve the free optimization problem
min L(x, û)
x∈IR n
instead of the constrained problem
min f(x)
s.t. g i (x) ≤ 0,
i =1, ···,m.