Page 124 -

Ax = b, x ≥ 0, then the first of Strang’s lemmas tell us
where
u − u
h V
F(x; p) = max d(p)y : W(p)y
≤ C(inf ( u − v
h V
v h ∈V h
= w(p) − T(p)x, y ≥ 0. |a(v ,w ) − a (v ,w )|
h
h
h
h
h
+ sup )
Here the second stage variable is denoted y.It is w h ∈V h w
h V
determined after x has been set and the random |f(w ) − f (w )|
h
h
h
variable p has been realized. The LP data depend + sup w h ∈V h ),
w
h V
on p as functions, d(p), W(p), w(p), and T(p).
with C = C(α, a , a )> 0. The second
The fixed recourse model has W(p) = W. The h
Strang’s lemma targets a non-conforming choice
complete recourse model assumes it is fixed and
m
{Wy : y ≥ 0} is all of R (where m = number of V , that is V ⊂ V .As . is not necessarily
h
h
V
well defined for functions of V , we have to intro-
of rows of W). This means that no matter what
duce a mesh-dependent norm . on V +V for
value of x is chosen for the first stage, there is h h
which a is elliptic
feasible recourse (y). This is simple recourse if h
W = [I − I], so we can think of y as having |R{a (v ,u )}| ≥ α v ∀v ∈ V .
two parts: y pos and y neg . The second stage LP h h h h h h h
simplifies to the following: Moreover, we have to require continuity
max d pos (p)y pos + d neg (p)y neg : |a (u, v )|≤ c u v
h h h h h
y pos ,y neg ≥ 0 ∀u ∈ V + V, v ∈ V .
h
h
h
y pos − y neg = w(p) − T(p)x. Then with C = C(α, c) > 0
The certainty equivalent depends upon the
u − u ≤ C(inf u − v
h V
h V
underlying decision process. If it is adaptive, the v h ∈V h
recourse model applies (but RO might be more |a(v ,w ) − a (v ,w )|
h
h
h
h
h
practical). The chance constraint model repre- + sup w h ∈V h ).
w
h V
sents a notion of an allowed frequency of viola-
tions, as in environmental control models. stratopause That region of the atmosphere
which lies between the stratosphere and the
stoichiometry The number of moles of a mesosphere and in which a maximum in the tem-
reactant used in a reaction, normalized to the perature occurs.
number of moles of all the other reactants.
Comment: Stoichiometries are always inte- stratosphere The atmospheric shell lying
gers, because they are determined by chemical just above the troposphere which is characterized
indivisibility of atoms. See mole. by an increasing temperature with altitude. The
stratosphere begins at the tropopause (about 10–
Strang’s lemmas Consider a linear vari-
15 km height) and extends to a height of about
ational problem a(u, v) = f (v), v ∈ V posed
50 km, where the lapse rate changes sign at the
over a Hilbert space V . Commiting a vari-
stratopause and the beginning of the mesosphere.
ational crime the corresponding discrete vari-
ational problem reads strict interior Let {x ∈ X : g(x) ≤ b} be
the level set of g. Then, its strict interior is {x ∈
u ∈ V : a (u ,v ) = f (v ) ∀v ∈ V ,
h h h h h h h h h
X : g(x)<b}. (This is not to be confused
where V ⊂ V is a V -conforming finite element with the relative interior or the set interior. See
h
space, a : V × V → C a sesqui-linear form, interior.)
h h h
and f : V → C a linear form. If a isV-elliptic,
h h h
that is, strictly complementary Each complemen-
tary pair of variables must have exactly one zero
2
|R{a (v ,u )}| ≥ α v ∀v ∈ V , (the other positive).
h h h h V h h
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC