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∗
where e> 0. The tangent plane at x is the where f (k) is the kth derivative of f . Truncating
set of all initial derivatives: {x (0)}. (This is the series at the nth term, the error is given by:

a misnomer, except in the special case of one
n (k) ,
,
function and two variables at a nonstationary , f (h) k ,
|E (h)|= ,f(x) − (x − h) ,.
n
point.) An important fact that underlies the clas- k!
k=0
sical Lagrange multiplier theorem when the rank
of grad h(x ) is full row (x is then called a reg- This is a Taylor expansion, and for the Taylor
∗
∗
∗
ular point): the tangent plane is {d : grad h(x ) seriestoequalthefunctionalvalue, itisnecessary
d = 0}. that the error term approaches zero for each n:
Extending this to allow inequalities, the
equivalent of the tangent plane for a regular point lim E (h) = 0.
n
h→0
(x ) is the set of directions that satisfy first-order
∗
conditions to be feasible: In any case, there exists y in the line segment
[x, x + h] such that
∗
{d : grad h(x )d = 0 and (n+1)
f (y) n+1
E (h) = (y − h) .
n
∗
grad g (x )d ≤ 0 for all i : g (x ) = 0}. (n + 1)!
∗
i i
Taylor theorem Let f : (a−h, a+h) → R
target analysis This is a metaheuristic to n+1
be in C . Then, for x in (a, a + h),
solve global optimization problems, notably
combinatorial optimization, using a learning f(x) = f(a) + [f (1) (a)][x − a] +
mechanism. In particular, consider a branch and
n
bound strategy with multiple criteria for branch ... + [f (n) (a)][(x − a) ]/n! + R (x, a),
n
selection. After solving training problems, hind-
where R (x, a), called the remainder, is given by
n
sight is used to eliminate dead paths on the search
the integral:
tree by changing the weights on the criteria: set
w> 0 such that wV ≤ 0 at node i with value x (x − t) n
i (n+1)
V , that begins a dead path, and wV > 0 at each f (t) dt.
i i a n!
node, i, on the path to the solution. If such
weights exist, they define a separating hyper- This extends to multivariate functions and
plane for the test problems. If such weights do is a cornerstone theorem in nonlinear program-
not exist, problems are partitioned into classes, ming. Unfortunately, it is often misapplied as
using a form of feature analysis, such that each an approximation by dropping the remainder,
class has such weights for those test problems in assuming that it goes to zero as x → a.
the class. After training is complete, and a new n
telegraph equation Let U ⊂ R be open
problem arrives, it is first classified, then those
and u : U ×R → R. The telegraph equation for
weights are used in the branch selection.
u is
u + du − u xx = 0.
t
tt
n
Taylor expansion For f in C , Taylor’s the-
orem is used by dropping the remainder term.
temperature inversion (in atmospheric chem-
The first-order expansion is f(x) = f(y) +
istry) A departure from the normal decrease
gradf (x)(x − y), and the second-order expan-
of temperature with increasing altitude. A
sion is f(x) = f(y) + gradf (x)(x − y) + temperature inversion may be produced, for
t
(x − y) H (x)(x − y)/2. example, by the movement of a warm air mass
f
over a cool one. Intense surface inversions may
Taylor series For a function, f , having all form over the land during nights with clear skies
order derivatives, the series and low winds due to the radiative loss of heat
from the surface of the earth. The temperature
∞ (k)
f (h)
k
(x − h) , increases as a function of height in this case.
k! Poor mixing of the pollutants generally occurs
k=0
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC