Page 122 -

stronger one may fulfill demand (or part thereof ) (No direction vector is sought if gradf(x) = 0;
for another beam (but not conversely).The manu- such algorithms stop when reaching a stationary
facture of each type of steel beam involves a point.)

2
fixed charge for its setup. In addition, there is Other norms, such as d = d Qd, where Q
a shipping cost proportional to the difference in is symmetric and positive definite, lead to other
the demanded strength and the actual strength, directions that are steepest relative to that norm.
and proportional to the quantity shipped. In particular, if Q = H (x), this yields the modi-
f
fied Newton method.
Let
Steiner problem Find a subgraph of a
N = number of varieties of strengths
graph, say G = [V ,E ], such that V contains
D(t) = demand for beam of strength s t ∗
( where s ≥ s ≥ ... ≥ s ) V (a specified subset of nodes), and Sum{c(e) :
2
1
N

x(t) = amount of beams of strength s t e ∈ E } is minimized. It is generally assumed
∗
manufactured c ≥ 0. When |V |= 2, this is the shortest path
problem. When |V |=|V |, this is the (min-
∗
p (x(t)) = manufacturing cost of x(t) units
t
of beam of strength s (including) imum) spanning tree problem.
t
fixed charge
step size A scalar (s) in an algorithm of the
y(t) = total excess of beams of strength
form: x = x + sd, where d is the direction
s , ..., s before fulfilling demand
1
t
D(t + 1), ...,D(N) vector. After d is chosen (nonzero), the step
size is specified. One step size selection rule
h (y(t)) = shipping cost( = c[s t + 1) − s ]
(
t
t
Min{y(t), D(t)}). is line search; another is a fixed sequence, like
s = 1/k.
k
Althought doesnotindextime, themathemat- stepwise reaction A chemical reaction with
ical program for this problem is of the same form at least one reaction intermediate and involving
as the production scheduling problem, using the at least two consecutive elementary reactions.
inventory balance equations to relate y and x.
This is valid because s ≥ s ≥ ... ≥ s implies
1 2 N stereochemical formula (stereoformula) A
y(t) can be used to fulfill demand D(t + 1) + three-dimensional view of a molecule either as
D(t + 2) + ... + D(N). (Also, note that here h
t such or in a projection.
is not a holding cost.)
sticking coefficient (in surface chemistry)
steepest ascent (descent, if minimizing) A The ratio of the rate of adsorption to the rate at

class of algorithms, where x = x + sd, such which the adsorptive strikes the total surface, i.e.,
that the direction vector d is chosen by maximiz- covered and uncovered. It is usually a function
ing the initial velocity of change, and the step of surface coverage, of temperature and of the
size (s) is chosen by line search. Generally used details of the surface structure of the adsorbent.
in the context of unconstrained optimization, the
n
mathematical program is Max{f(x) : x ∈ R }, stiffness matrix The Galerkin discretiza-
1
where f is in C .(For descent, change Max to tion of a linear variational problem by means of a
Min.) Then, d is chosen to maximize the first- finite element space V leads to a linear system of
h
order Taylor approximation, subject to a normal- equations. If a stands for the sesqui-linear form
ization constraint: Max{gradf(x)d : d = 1}, of the variational problem, the system matrix is
where d denotes the norm of the direction vec- given by
tor, d. When the Euclidean norm is used, this
yields the original steepest ascent algorithm by A := (a(b ,b )) N ,
i j i,j=1
Cauchy, which moves in the direction of the gra-
dient: where {b , ··· ,b },N = dimV , is the nodal
h
1
N
basis of V . If the sesqui-linear form a arises
h
d = gradf(x)/ gradf(x) . from the weak formulation of a boundary value
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC