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 Uspekhi Mat. Nauk, 1968, Volume 23, Issue 6(144), Pages 51–116 (Mi umn5684)

Duality of convex functions and extremum problems

A. D. Ioffe, V. M. Tikhomirov

Abstract: Let $\mathfrak{X}$ be a real linear topological space and $\mathfrak{Y}$ its conjugate. We denote by $\langle x,y\rangle$ the value of the linear functional $y\in\mathfrak{Y}$ on the element $x\in\mathfrak{X}$. For real functions $f(x)$ on $\mathfrak{X}$ we introduce two operations: the ordinary sum
$$f_1(x)+f_2(x)$$
and the convolution
$$f_1\oplus f_2(x)=\inf_{x_1+x_2=x}(f_1(x_1)+f_2(x_2)),$$
and also the transformation associating with $f(x)$ its dual function on $\mathfrak{Y}$ which is obtained from $f(x)$ by the formula
$$f^*(y)=\sup_{x\in\mathfrak{X}}(\langle x,y\rangle-f(x)).$$
The following propositions hold.
1) The operation $^*$ is involutory:
$$f^{**}=f$$
if and only if $f(x)$ is a convex function and lower semicontinuous on $\mathfrak{X}$.
2) $(f_1\oplus f_2)^*=f_1^*+f_2^*$.
$$(f_1+f_2)^*=f_1^*\oplus f_2^*.$$
These theorems were proved for a finite-dimensional space by Fenchel [93] and in the general case by Moreau [60].
Chapter I is concerned with proving these theorems and generalizations of them.
Chapter II is concerned with their application to mathematical programming and the calculus of variations. Proofs are given of very general duality theorems of mathematical programming and saddle point theorems. Constructions are then given which lead to extensions of optimal control problems, and an existence theorem is proved for these problems.
Chapter III contains an investigation of problems of approximating $x\in\mathfrak{X}$ and the set $C\subset\mathfrak{X}$ by an approximating set $A\subset\mathfrak{X}$ using methods of the theory of duality of convex functions. Duality theorems for some geometric characteristics of sets in $\mathfrak{X}$ are derived at the end of the chapter.

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English version:
Russian Mathematical Surveys, 1968, 23:6, 53–124

Bibliographic databases:

UDC: 517.51+519.3+519.95
MSC: 46A20, 46A03, 46A55, 52A40, 51M16

Citation: A. D. Ioffe, V. M. Tikhomirov, “Duality of convex functions and extremum problems”, Uspekhi Mat. Nauk, 23:6(144) (1968), 51–116; Russian Math. Surveys, 23:6 (1968), 53–124

Citation in format AMSBIB
\Bibitem{IofTik68} \by A.~D.~Ioffe, V.~M.~Tikhomirov \paper Duality of convex functions and extremum problems \jour Uspekhi Mat. Nauk \yr 1968 \vol 23 \issue 6(144) \pages 51--116 \mathnet{http://mi.mathnet.ru/umn5684} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=288601} \zmath{https://zbmath.org/?q=an:0167.42202|0191.13101} \transl \jour Russian Math. Surveys \yr 1968 \vol 23 \issue 6 \pages 53--124 \crossref{https://doi.org/10.1070/RM1968v023n06ABEH001251} 

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