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Then, the approximating LP is: sequential quadratic programming (SQP)
Solving a nonlinear program by a sequence of

Max Df (k, j)u(k, j) : 0 ≤ u ≤ 1, quadratic approximations and using quadratic
kj
programming to solve each one. The approxima-
tions are usually done by using the second-order
Dg(k, j)u(k, j) ≤ b,
kj Taylor expansion.

Dh(k, j)u(k, j) = c,
kj sequential unconstrained minimization tech-
nique (SUMT) This is the penalty function

where b = − g (x(0, j)) and c = − h
j j j j approach.
(x(0, j)) (a similar constant was dropped from
the objective). Another restricted basis rule is
sesqui-linear Describing a complex-valued
invoked: u(k, j) > 0 implies u(k, q) = 1 for
function of two variables which is linear in the
all q < j and all k.
first variable and conjugate-linear in the second.
separating hyperplane A hyperplane for
series A formal sum a , where < a >
j
which two (given) sets lie in opposite half spaces. j j
is a sequence.
The separation is strict if the two sets are con-
tained in their respective open half space.
set An unordered collection of elements,
without duplicates, each of which elements satis-
sequence An ordered countable collection
fies some property. An enumerated set is delim-
of elements which can include duplicates. The
ited by braces ({x}).
ordering need not reflect a mathematical relation.
Comment: Some authors permit sets to
An enumerated sequence is delimited by angle
include duplicates. See also bag, list, sequence,
brackets (/x0).
and tuple.
Comment: The key notion of a sequence is
that of succession: that for any two elements of a
sequence, a and b, a ≺ b iff a occurs to the left of set difference Given two sets, A and B, their
b in the sequence (we adopt the convention that difference, D = A − B, is the set of all elements
we read the sequence from left to right). Be care- of A not found in B: ∀d ∈ D, d ∈ A and d ∈ B.
ful not to confuse ≺ with <; a could be 12 and b Comment: Synonymous with set subtraction.
could be 5, yet still a ≺ b (read “a precedes b”). Similar operations can be defined for bags, lists,
Classic examples of sequences are paths through and sequences. See also symmetric difference.
graphs and strings, such as DNA sequences.
set of reactions A collection of reactions.
sequencing problems Finding an ordering,
or permutation, of a finite collection of objects, set subtraction See set difference.
like jobs, that satisfies certain conditions, such as
precedence constraints. shadow price An economic term to denote
the rate at which the optimal value changes with
sequential decision process See time- respect to a change in some right-hand side that
staged. represents a resource supply or demand require-
ment. This is sometimes taken as synonymous
sequential linear programming (SLP) with the dual price, but this can be erroneous, as
Solving a nonlinear program by a sequence in the presence of degeneracy.
of linear approximations and using linear
programming to solve each one. The linear shape function Given a finite element
approximations are usually done by using the (K, V ,X ) the set of local shape functions is
K
K
first-order Taylor expansion. the basis of V that is dual to X .
K K

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