Nonstandard Models and Optimization

S. S. Kutateladze


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.


The equation
with X and Y Banach spaces and T:X→Y, is approximated with the equation
TN xN=yN
on choosing finite-dimensional approximating subspaces XN and YN and the embeddings iN,jN:


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:=infx ∈ Cf(x) (if existent) is called the value of (C,f). A feasible element x0 is an ideal optimum or solution provided that e=f(x0). In other words, x0 is an ideal optimum if and only if f(x0) is the least element of the image f(C); i. e., x0 ∈ C and f(C) ⊂ f(x0)+E+.


Fix a positive element ε∈ E. A feasible point x0 is an ε-solution or ε-optimum of a program (C,f) provided that f(x0) ≤ e+ε with e the value of (C,f). In other words, x0 is an ε-solution of (C,f) if and only if x0 ∈ C and the f(x0)-ε is the greatest lower bound of f(C) or, equivalently, f(C)+ε ⊂ f(x0)+E+. Clearly, x0 is an ε-solution of an unconditional problem f(x)→ inf if and only if the zero belong to ∂ε f(x0); i. e.,
f(x0) ≤
x ∈ X 
 f(x)+ε  ↔ 0 ∈ ∂ε f(x0).
Here ∂ε f(x0) is the ε-subdifferential of f at x0. Recall that each member of ∂ε f(x0) is a linear operator from X to E such that (∀x ∈ X) l(x-x0)≤ f(x)-f(x0)+ε.


A feasible point x0 is ε-Pareto optimal for (C,f) whenever f(x0) is a minimal element of U+ε, with U:=f(C); i. e., (f(x0)-E+)∩(f(C)+ε)=[f(x0)]. In more detail, x0 is ε-Pareto-optimal means that x0 ∈ C and, for all x ∈ C, from f(x0) ≥ f(x)+ε it follows that f(x0) = 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.


[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.

April 22, 2008

This talk was prepared for the International Conference “Methods of Logic in Mathematics V” at St. Petersburg June 1–6, 2008, but undelivered due to illness. See slides in PDF. A complete version is avalable as arXiv:0805.3947 and Vladikavkaz. Math. Jour., 2008, V. 10, No. 4, 39–48.

File translated from TEX by TTH, version 3.78.
On 23 Apr 2008, 06:06.
English Page
Russian Page