At the end of the 1940s Leonid Kantorovich
formulated the thesis of interdependence between
functional analysis and applied mathematics. He distinguished
the Cauchy method of majorants, the method of finite-dimensional approximations,
and the Lagrange method for the new optimization problems motivated
by economics.
Let X and Y be real vector spaces endowed with
lattice norms | ·|_{X}
and | ·|_{Y}. Assume that T is a linear operator
from X to Y, and S is a positive operator from
E to F satisfying
In case
| Tx |_{Y} ≤ S| x|_{X} (x ∈ X),
we call S a dominant of T.
The technique of model theory demonstrated that
the main properties of lattice-normed spaces are Boolean interpretations
of the properties of classical normed spaces.
Each Banach space inside a Boolean valued universe deciphers
into a universally complete lattice normed space.
Moreover, each lattice normed space may be presented as
a dense subspace of a Banach space in an appropriated Boolean valued model.
Nonstandard models yield the method of hyperapproximation
Here E and F are normed spaces over the same
scalars, while
T is a bounded linear operator from E to F, and
^{#} symbolizes a nonstandard hull.
Let X be a real vector space and let
E be an ordered vector space. Assume that
f:X→E^{·}:=E∪+∞ is a convex operator
and C ⊂X is a convex set.
A vector program is a pair
(C,f) written as
x ∈ C, f(x)→
inf
.
The element
e:=inf_{x ∈ C}f(x)
(if existent) is called the
value of (C,f).
A feasible element x_{0} is an
ideal optimum or solution provided that e=f(x_{0}).
In other words,
x_{0} is an ideal optimum if and only if
f(x_{0}) is the least element of the image f(C);
i. e., x_{0} ∈ C and
f(C) ⊂ f(x_{0})+E^{+}.
Fix a positive element ε∈ E.
A feasible point x_{0} is an
ε-solution
or ε-optimum
of a program (C,f) provided that f(x_{0}) ≤ e+ε with
e the value of (C,f). In other words, x_{0}
is an ε-solution of (C,f)
if and only if x_{0} ∈ C and the
f(x_{0})-ε is the greatest
lower bound of f(C) or, equivalently,
f(C)+ε ⊂ f(x_{0})+E^{+}.
Clearly, x_{0} is an ε-solution
of an unconditional problem f(x)→ inf if and only if
the zero belong to ∂_{ε} f(x_{0}); i. e.,
f(x_{0}) ≤
inf
x ∈ X
f(x)+ε ↔ 0 ∈ ∂_{ε} f(x_{0}).
Here
∂_{ε} f(x_{0}) is the ε-subdifferential
of f at x_{0}. Recall that each member
of ∂_{ε} f(x_{0}) is
a linear operator from X to E such that (∀x ∈ X)
l(x-x_{0})≤ f(x)-f(x_{0})+ε.
A feasible point x_{0} is
ε-Pareto optimal for (C,f)
whenever f(x_{0}) is a minimal element of U+ε,
with U:=f(C); i. e.,
(f(x_{0})-E^{+})∩(f(C)+ε)=[f(x_{0})].
In more detail, x_{0} is ε-Pareto-optimal
means that x_{0} ∈ C and, for all x ∈ C,
from f(x_{0}) ≥ f(x)+ε it follows that
f(x_{0}) = f(x)+ε.
Every ε-solution with a sufficiently small
ε may be viewed as a candidate for the position of
a “practical optimum,”a solution suitable for practical usage.
The calculus rules for
ε-subdifferentials bring about a formal machinery
to account for the accuracy of a solution of the extremal problem under
consideration.The relevant technique is rather elaborate,
subtle, and attractive nowadays.
However, the exact formulas are often bulky and do not
agree fully with the practical tricks of optimization
when we use the simplified rules of “neglecting minor errors.”
An adequate technique free of these shortcomings is
due to the modern opportunities of nonstandard set theory.
References
[1] Kantorovich L. V. (1948)
“Functional Analysis and Applied Mathematics,”
Vestnik LGU 6, 3–18 [in Russian].
[2] Kusraev A. G. and Kutateladze S. S. (2005)
Introduction to Boolean Valued Analysis.
Moscow: Nauka Publishers [in Russian].
[3] Kusraev A. G. and Kutateladze S. S. (2007).
Subdifferential Calculus: Theory and Applications.
Moscow: Nauka Publishers [in Russian].
[4] Liu J. C. (1999)
“ε-Properly efficient solutions to nondifferentiable
multiobjective programming problems,” Appl. Math. Lett.
12:6, 109–113.
[5] Gutiérrez C., Jiménez B., Novo V. (2006)
“On approximate
solutions in vector optimization problems via scalarization,”
Computational Optimization and Applications, 35:3, 305–324.
[6] Gutiérrez C., Jiménez B., Novo V. (2007)
“Optimality conditions for metrically consistent
approximate solutions in vector optimization,”
J. Optim. Theory Appl., 133:1, 49–64.