Some of the ideas are briefly as follows. The
minimization of a function f(x), subject to constraints
g(x) <= 0, can be described using a Lagrangian function
f(x) + vg(x), where v is a Lagrange multiplier. If f is
vector valued, then "minimum" at a point p must be
redefined, replacing f(x) >= f(p) by NOT [f(x) < f(p)],
with some ordering of the vectors. The Lagrangian L is
replaced by tf(x) + vg(x), with an additional multiplier
t. If the functions are differentiable, then L has zero
gradient at the minimum, for some t and v. There are
analogs of this result when the functions are not
differentiable, or their values are sets instead of
points. One approach is to smooth f and g by averaging
their values over nearby points; this gives a smooth
problem, approximating the given one.
When do the Lagrangian conditions, in turn, imply a
minimum? This is true if the vector function F = (f,g) is convex,
thus if F(x) - F(p) >= F'(p)(x-p). But we can replace x-p by
some "scale function" of x-p, and it still works. This
makes F an "invex" vector function. This extends to
functions F that are not differentiable, in two ways.
One can replace derivatives by tangent cones to the graph
of F. Or F may satisfy an inequality
tF(x) + (1-t)F(p) >= F(z) for some suitable z,
depending on x,p,t. We still get sufficient conditions.
If there is an equality constraint h(x) = 0, then this
makes x fall on some curved surface. More generally,
what happens to "invex" when x lies on a manifold? A
somewhat restricted "invex" corresponds to
"convexifiable", thus F can be made convex by
transforming the underlying space. This idea works for
manifolds, if we assume singular points are well behaved,
and then "invex" corresponds to a topological property
("zero index") at singular points. Extensions to vector
functions are being sought.
Lagrange multipliers, like shadow prices in linear
programming, measure the sensitivity of a minimum value
to small perturbations of the problem. It is harder to
find what happens to a minimum point when the problem is
perturbed. This requires the problem to have a stability
property, and this happens when "invex" is strengthened a
bit. All this works also with optimal control problems,
where x is replaced by a "state function" and a "control
function".
Optimal control questions are equivalent to mathematical
programs in infinite-dimensional spaces, and Lagrangian
conditions apply. The Pontryagin theory can be deduced
from this viewpoint (see my 1995 book), and this has led to
the development of two computer packages for optimal control
(OCIM and SCOM); the latter, using MATLAB, runs on a
desktop computer, and facilitates investigations, such as
sensitivity analysis, witha minimum of programming. A
number of optimal control models for economics (especially
growth models) and finance have been developed (with Islam),
and some of them computed.
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