Labeling algorithm. A class of algorithms for path-related
problems on graphs and
networks. To illustrate, consider a
network [V,A] with arc values c.
Dijkstra's
shortest path algorithm (c = distance).
- Input. source (s), destinations (T not Ø),
distances (c).
- Output. shortest path from s to each node in T.
- Initialize labels: L(v)=inf for v not= s and L(s)=0.
Set S=Ø.
- Label: select u in argmin{L(v): v in V\S}
(terminate if S=V). Then, update S := S\/{u}, and for v in V\S:
if L(u) + c(u,v) < L(v), set L(v) := L(u) + c(u,v).
Terminate when T is a subset of S (i.e., all destination nodes
are labeled with their shortest distance from node s).
A generic
shortest path labeling algorithm from all nodes to destination
node t (c = distance).
- Input. destination (t), distances (c).
- Output. shortest path from each node to t.
- Initialize labels, L(v) = estimate of shortest path length
from node v to node t, with L(t) = 0.
- Label: Find an arc, say (i, j), that violates the distance
equations, say L(j) > L(i) + c(i, j). If none, terminate;
otherwise, update the label: L(j) = L(i) + c(i, j).
The labeling step is repeated until there are no violations.
At termination, L(j) = length of shortest path from node j to
node t.
Another type of labeling algorithm occurs in
simplicial subdivision,
used to compute a
fixed point that represents an economic equilibrium.
Lagrange conditions. The optimality equation (plus
feasibility conditions) in
Lagrange's multiplier theorem.
Here is
biography of Lagrange.
Lagrange Multiplier Rule (LMR). From the extension to
Lagrange's multiplier theorem:
Suppose x* is in argmax{f(x): g(x) <= 0, h(x) = 0},
where f, g, h are in C^1. Then, there exist multipliers
(u, v) for which the following conditions hold:
- grad_x_[f(x*) - ug(x*) - vh(x*)] = 0;
- u >= 0;
- ug(x*)=0.
Since g(x*) <= 0, the last condition, given u >= 0, is equivalent to
complementary slackness.
These are considered
first-order optimality conditions, though the Lagrange Multiplier
Rule is not always valid -- see
constraint qualifications.
Lagrange's multiplier theorem. Let f, h be in C^1 and suppose
grad_h(x*) has full row rank. Then, x* is in argmax{f(x): h(x)=0}
only if there exists v in R^m such that:
grad_f(x*) - v grad_h(x*) = 0.
v_i is called the Lagrange multiplier associated with the i-th
constraint (h_i(x)=0). Extensions to remove the rank condition and/or allow
inequality constraints were by
Carathéodory,
John, Karush, and
Kuhn & Tucker. Also see the
Lagrange Multiplier Rule.
Lagrangian. For the mathematical program in
standard form, the Lagrangian is the
function:
L(x, u, v) = f(x) - u g(x) - v h(x) for x in X and u >= 0.
Note that the