Page 114 -

second-order conditions A descendant self-concordance Properties of a function
from classical optimization, using the that yield nice performance of Newton’s method
second-order term in Taylor’s expansion. used for line search when optimizing a barrier
For unconstrained optimization, the second- function. Specifically, let B be a barrier function
2
order necessary condition (for f in C )is for S ={x ∈ X : g(x) ≤ 0} with strict interior
0
that the Hessian is negative semidefinite (for a S . Let x be in S and let d be a direction vector
n
max). Second-order sufficient conditions are the in R such that the line segment [x − td, x + td]
∗
∗
first-order conditions plus the requirement that is in S for t in [0,t ], where t > 0. Then, define
∗
the Hessian be negative definite. F :[0,t ] −→ R by:
For constrained optimization, the second-
order conditions are similar, using projection F(t) = B(x + td)
for a regular mathematical program and the
(while noting that F depends on x and d). The
Lagrange multiplier rule. They are as follows
2
(all functions are in C , and the mathematical function F is self-concordant if it is convex in
3
C and satisfies the following for all x and d:
∗
program is in standard form, for x a local maxi-
mum): (3/2)
|F (0)|≤ 2F (0) .
(i.) Second-order necessary conditions.
There exist Lagrange multipliers, (u, v), such One calls Fk-self-concordant in an open convex
that u ≥ 0 and ug(x ) = 0 for which: (1) domain if
∗
grad [L(x ,u,v)] = 0, and (2) H [L(x ,u,v)] (3/2)
∗
∗
x
x
is negative semi-definite on the tangent plane. |F (0)|≤ 2kF (0) .
(ii.) Second-order sufficient conditions. The
The logarithmic barrier function, associated with
above necessary conditions hold but with (2)
linear programming, is self-concordant with k =

replaced by (2 )H [L(x ,u,v)] is negative def- 1. This further extends naturally to functions
∗
x
inite on the tangent plane. n
in R .
selectively labeled An isotopically labeled
semantic mapping A bijective, partial func-
compound is designated as selectively labeled
tion between each member of the set of symbols,
when a mixture of isotopically substituted com- σ ∈ K, and its semantics: ω: σ −→ ω(σ ),
j
j
j
pounds is formally added to the analogous ω the mapping operator. The set of semantic
isotopically unmodified compound in such a way mappings, ;, is defined for each element of the
that the position(s) but not necessarily the num- domain and codomain to which they apply.
ber of each labeling nuclide is defined. A selec- Comment: Notice that ω is a partial func-
tively labeled compound may be considered as a tion. There will be elements of the domain (the
mixture of specifically labeled compounds. symbol set of the language) for which a given
mapping will not be defined. See also semantics
self-adjoint operator A linear operator T and semiote.
on a Hilbert space (H,<,>) such that T ∗ =
T , where the (Hilbert)-adjoint T is defined by semantics For each symbol σ in the alpha-
∗
j
∗
<x, Ty >=<T x, y >, x, y ∈ H. See also bet K, j a positive integer index, its semantics,
symmetric operator. A symmetric operator T is ω(σ ), is a computationally executable definition
j
called essentially self-adjoint if its closure T is of the meaning of σ . It is found or produced by
¯
j
self-adjoint. applying a semantic mapping ω to σ , denoted
j
ω: σ −→ ω(σ ), such that ω is one-to-one,
j
j
self-avoiding random walk A random walk onto, and defined for that σ . Under these condi-
j
which does not pass any space point twice. In tions, we call both symbol and mapping seman-
three dimensions, this is a more realistic model tically well formed.
for polymer chains. See Gaussian chain and Comment: This is equivalent to saying that
excluded volume. every term in a language, L, which describes
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC