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strictly concave function Negative of a subbag See parts of collections.
strictly convex function.
subdifferential (of f at x) ∂ (x) ={y : x
f
strictly convex function A convex function is in argmax {v − f(v) : v ∈ X}}.If f
y
that also satisfies the defining inequality strictly is convex and differentiable with gradient,
for distinct points, say x and y: gradf, ∂ (x) ={gradf(x)}. Example: f(x) =
f
|x|. Then,∂ (0) = [−1, 1].
f
f(ax + (1 − a)y)<af (x) + (1 − a)f (y) The subdifferential is built on the concept of
supporting hyperplane, generally used in convex
for all a in (0, 1). analysis. When f is differentiable in a deleted
neighborhood of x (but not necessarily at x), the
strictly quasiconcave function Negative of B-subdifferential is the set of limit points:
a strictly quasiconvex function.
k
∂ f(x) ={d : there exists {x } >x
B
strictly quasiconvex function X is a convex
k
set and f(ax + (1 − a)y) < Max {f (x), f (y)} and {gradf(x )} >d}.
for all x, y in X for which f(x) = f(y), and a
If f is continuously differentiable in a
is in (0, 1). Note: f need not be quasiconvex.
neighborhood of x (including x), ∂ f(x) =
B
{gradf(x)}. Otherwise, ∂ f(x) is generally not
B
strong collision A collision between two
a convex set. For example, if f(x) =|x|,
molecules in which the amount of energy trans-
∂ f(0) ={−1, 1}.
B
ferred from one to the other is large compared
The Clarke subdifferential is the convex hull
with k T , where k is the Boltzmann constant of ∂ f(x).
B
B
and T the absolute temperature. B
subgradient A member of the subdifferen-
strongly concave function Negative of a
tial.
strongly convex function.
subgraph A graph G (V , E ) is a subgraph

strongly convex function A function f in of G(V, E) if every node and edge present in G
2
C with eigenvalues of its Hessian bounded away
is present in G; that is, V ⊆ V and E ⊆ E. G
from zero (from below): there exists K> 0
2 is a proper subgraph of G if G = G.
such that h H (x)h ≥ K h for all h in
f
n
R . For example, the function 1 − exp(−x)
sublist See parts of collections.
is strictly convex on R, but its second derivative
is − exp(−x), which is not bounded away from
submodular function Let N be a finite set
zero. The minimum is not achieved because the
and let f be a function on the subsets of N into
function approaches its infimum of zero without
R. Then, f is submodular if it satisfies:
achieving it for any (finite) x. Strong convexity
rules out such asymptotes. f(S ∧ T) ≤ f(S) + f(T ) − f(S ∧ T)
strongly quasiconcave function Negative for S, T subsets of N.
of a strongly quasiconvex function.

subnetwork A network N (V , E , P , L ) is

strongly quasiconvex function (f on X ) a subnetwork of N(V, E, P, L) if every node,
On a convex set X f (ax + (1 − a)y) < Max edge, parameter, and label present in N is present

{f (x), f (y)} for all x, y in X, with x = y, and in N; that is, V ⊆ V; E ⊆ E; P ⊆ P; and

a in (0, 1). L ⊆ L. N is a proper subnetwork of N if also

N = N.

subadditive function f(x + y) ≤ f(x) +
f(y) where x, y in the domain implies x + y is subsequence A subset of a sequence, with
in the domain. the order preserved.
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC