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tangent cone Let S be a subset of R and
let x be in S. The tangent cone, T(S, x ),is
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∗
the set of points y such that there exist sequences
T {a },a ≥ 0 and {x } in S such that {x }→ x ∗
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n
n
n
n
and { a (x −x )−y } → 0. This arises in con-
∗
n
nection with the Lagrange multiplier rule much
like the tangent plane, though it allows for more
general constraints, e.g., set constraints. In par-
tableau (pl. tableaux) A detached coef-
ticular, when there are only equality constraints,
ficient form of a system of equations, which can
h(x) = 0,T (S, x ) = null space of grad h(x )
∗
∗

change from x + Ay = b to x + A y = b .

if grad h(x ) has full row rank. (There are some
∗
The primes denote changes caused by multiply-
subtleties that render the tangent cone more gen-
ing the first equation system by the basis inverse
eral, in some sense, than the tangent plane or
(a sequence of pivots in the simplex method).
null space. It is used in establishing a necessary
Other information could be appended, such as
constraint qualification.)
the original bound values.
tangent lift A general procedure to associate
tabu search This is a metaheuristic to solve canonically an object on the tangent bundle TM
global optimization problems, notably combina- of a manifold M once an object is given on M.
torial optimization, based on multilevel mem- In particular:
ory management and response exploration. It
(i.) Tangent lift of a parametrized curve γ :
requires the concept of a neighborhood for a trial dγ
0
solution (perhaps partial). In its simplest form, a I ⊂ R → M: let us define τ (t ) = dt | t=t 0
γ
the tangent vector to the curve γ at t = t ∈ I.
tabu search appears as follows: 0
We can define the tangent lift of γ as the curve
µ
(i.) Initialize. Select x and setTabu List T = ˆ γ : I → TM : t → (γ (t), τ (t)).If γ (t) is
γ
µ
∗
null. If x is feasible, set x = x and f = f(x ); the local expression of γ , then (γ (t), ˙γ(t)) is
∗
∗
otherwise, set f =− inf (for minimization set the local expression of ˆγ .
∗
∗
f = inf). (ii.) Tangent lift of a map φ : M → N:if
(ii.) Select move. Let S(x) = set of neigh- tangent vectors are identified with derivations of
bors of x.If S(x)\T is empty, go to update. the algebra of local functions, the tangent map
Otherwise, select y in argmax {E(v) : v in Tφ : TM → TN : v → w is defined by
µ
S(x)\T }, where E is an evaluator function that w(f ) = v(f ◦ φ).If x µ = φ (x) is the local
measures the merit of a point (need not be the expression of φ, then the local expression of the
original objective function, f ). If y is feasible tangent map Tφ is given by:
∗
∗
∗
∗
and f (y)>f , set x = y and f = f(x ).
µ
x µ = φ (x)
Set x = y (i.e., move to the new point).
ν
µ
w µ = v ∂ φ (x)K
(iii.) Update. If some stopping rule holds, ν
stop. Otherwise, update T (by some tabu update
(iii.) Tangent lift of a vector field ξ =
rule) and return to select move. ξ (x) ∂ ∈ X(M): a vector field ˆ ξ
µ
µ =
µ
µ
There are many variations, such as aspiration ξ (x) ∂x ∂ µ + ˆ ξ (x) ∂v ∂ µ over TM locally given
µ
ν
µ
levels, that can be included in more complex by ˆ ξ = ξ ∂ ξ . Notice that the tangent lift of
ν
specifications. a commutator coincides with the commutator of
ˆ
lifts, i.e., [ξ, ζ] = [ ˆ ξ, ˆ ζ].
tangent (function) The function
tangent plane Consider the surface, S =
1
n
sin x {x ∈ R : h(x) = 0}, where h is in C .A
∗
tan(x) = . differentiable curve passing through x in S is
cos x
{x(t) : x(0) = x and h(x(t)) = 0 for all t in
∗
See cosine, sine. (−e, e)}, for which the derivative, x (t), exists,

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© 2003 by CRC Press LLC
© 2003 by CRC Press LLC