RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive
Impact factor
Subscription
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Uspekhi Mat. Nauk:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

This article is cited in 43 scientific papers (total in 43 papers)

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) $(f_1\oplus f_2)^*=f_1^*+f_2^*$.
3) Under certain additional assumptions
$$ (f_1+f_2)^*=f_1^*\oplus f_2^*. $$
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.

Full text: PDF file (6124 kB)
References: PDF file   HTML file

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}


Linking options:
  • http://mi.mathnet.ru/eng/umn5684
  • http://mi.mathnet.ru/eng/umn/v23/i6/p51

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. M. Tikhomirov, “Best methods of approximation and interpolation of differentiable functions in the space $C_{[-1,1]}$”, Math. USSR-Sb., 9:2 (1969), 275–289  mathnet  crossref  mathscinet  zmath
    2. A. D. Ioffe, “Subdifferentsialy ogranichenii vypuklykh funktsii”, UMN, 25:4(154) (1970), 181–182  mathnet  mathscinet  zmath
    3. L. V. Kantorovich, “Methods of optimization and mathematical models in economics”, Russian Math. Surveys, 25:5 (1970), 105–107  mathnet  crossref  mathscinet  zmath
    4. A. M. Vershik, “Some remarks on the infinite-dimensional problems of linear programming”, Russian Math. Surveys, 25:5 (1970), 117–124  mathnet  crossref  mathscinet  zmath
    5. V. L. Makarov, A. M. Rubinov, “Superlinear point-set maps and models of economic dynamics”, Russian Math. Surveys, 25:5 (1970), 125–169  mathnet  crossref  mathscinet  zmath
    6. G. Sh. Rubinshtein, “Duality in mathematical programming and some problems of convex analysis”, Russian Math. Surveys, 25:5 (1970), 171–200  mathnet  crossref  mathscinet  zmath
    7. V. D. Milman, “Geometric theory of Banach spaces. Part II. Geometry of the unit sphere”, Russian Math. Surveys, 26:6 (1971), 79–163  mathnet  crossref  mathscinet  zmath
    8. V. I. Arkin, V. L. Levin, “Convexity of values of vector integrals, theorems on measurable choice and variational problems”, Russian Math. Surveys, 27:3 (1972), 21–85  mathnet  crossref  mathscinet  zmath
    9. S. S. Kutateladze, A. M. Rubinov, “Minkowski duality and its applications”, Russian Math. Surveys, 27:3 (1972), 137–191  mathnet  crossref  mathscinet  zmath
    10. N. P. Korneichuk, “Inequalities for differentiable periodic functions and best approximation of one class of functions by another”, Math. USSR-Izv., 6:2 (1972), 417–428  mathnet  crossref  mathscinet  zmath
    11. A. D. Ioffe, “Convex functions occurring in variational problems and the absolute minimum problem”, Math. USSR-Sb., 17:2 (1972), 191–208  mathnet  crossref  mathscinet  zmath
    12. N. P. Korneichuk, “On extremal problems in the theory of best approximation”, Russian Math. Surveys, 29:3 (1974), 7–43  mathnet  crossref  mathscinet  zmath
    13. R. S. Ismagilov, “Diameters of sets in normed linear spaces and the approximation of functions by trigonometric polynomials”, Russian Math. Surveys, 29:3 (1974), 169–186  mathnet  crossref  mathscinet  zmath
    14. V. L. Levin, “Convex integral functionals and the theory of lifting”, Russian Math. Surveys, 30:2 (1975), 119–184  mathnet  crossref  mathscinet  zmath
    15. V. E. Maiorov, “Diskretizatsiya zadachi o poperechnikakh”, UMN, 30:6(186) (1975), 179–180  mathnet  mathscinet  zmath
    16. Daniel McFadden, “Tchebyscheff bounds for the space of agent characteristics”, Journal of Mathematical Economics, 2:2 (1975), 225  crossref
    17. M. A. Krasnoselskii, A. V. Pokrovskii, “O razryvnom operatore superpozitsii”, UMN, 32:1(193) (1977), 169–170  mathnet  mathscinet  zmath
    18. B. Sh. Mordukhovich, “Approksimatsiya i printsip maksimuma v negladkikh zadachakh optimalnogo upravleniya”, UMN, 32:4(196) (1977), 263–264  mathnet  mathscinet  zmath
    19. E. S. Levitin, A. A. Milyutin, N. P. Osmolovskii, “Conditions of high order for a local minimum in problems with constraints”, Russian Math. Surveys, 33:6 (1978), 97–168  mathnet  crossref  mathscinet  zmath
    20. Allan Pinkus, “Matrices and n-Widths”, Linear Algebra and its Applications, 27 (1979), 245  crossref
    21. V. I. Berdyshev, “Continuity of a multivalued mapping connected with the problem of minimizing a functional”, Math. USSR-Izv., 16:3 (1981), 431–456  mathnet  crossref  mathscinet  zmath  isi
    22. Gluskin E., “on Some Finite Dimensional Problems of the Theory of Widths”, no. 3, 1981, 5–10  isi
    23. Robert Whitley, “Markov and Bernstein's inequalities, and compact and strictly singular operators”, Journal of Approximation Theory, 34:3 (1982), 277  crossref
    24. N. P. Korneichuk, “S. M. Nikol'skii and the development of research on approximation theory in the USSR”, Russian Math. Surveys, 40:5 (1985), 83–156  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    25. I. I. Dzhioeva, “On some duality formulae for differential forms on compact Riemannian manifolds”, Russian Math. Surveys, 40:6 (1985), 121–122  mathnet  crossref  mathscinet  adsnasa
    26. A. Ya. Azimov, “Duality of multiobjective problems”, Math. USSR-Sb., 59:2 (1988), 515–531  mathnet  crossref  mathscinet  zmath
    27. Hidefumi Kawasaki, “An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems”, Math Program, 41:1-3 (1988), 73  crossref  mathscinet  zmath  isi
    28. S.L Zabell, “Mosco convergence in locally convex spaces”, Journal of Functional Analysis, 110:1 (1992), 226  crossref
    29. A. Jourani, “Regularity and strong sufficient optimality conditions in differentiable optimization problems”, Numerical Functional Analysis and Optimization, 14:1-2 (1993), 69  crossref
    30. A. Jourani, “Metric regularity and second-order necessary optimality conditions for minimization problems under inclusion constraints”, J Optim Theory Appl, 81:1 (1994), 97  crossref  mathscinet  zmath  isi
    31. Farhad Hüsseinov, “Interpretation of Aubin's fuzzy coalitions and their extension”, Journal of Mathematical Economics, 23:5 (1994), 499  crossref
    32. S. N. Kudryavtsev, “Diameters of classes of smooth functions”, Izv. Math., 59:4 (1995), 741–764  mathnet  crossref  mathscinet  zmath  isi
    33. V. S. Klimov, “Topological characteristics of non-smooth functionals”, Izv. Math., 62:5 (1998), 969–984  mathnet  crossref  crossref  mathscinet  zmath  isi
    34. A. I. Kozko, A. V. Rozhdestvenskii, “On Jackson's Inequality for Generalized Moduli of Continuity”, Math. Notes, 73:5 (2003), 736–741  mathnet  crossref  crossref  mathscinet  zmath  isi
    35. A. I. Kozko, A. V. Rozhdestvenskii, “On Jackson's inequality for a generalized modulus of continuity in $L_2$”, Sb. Math., 195:8 (2004), 1073–1115  mathnet  crossref  crossref  mathscinet  zmath  isi
    36. E. M. Skorikov, “The information Kolmogorov width and some exact inequalities between widths”, Izv. Math., 71:3 (2007), 603–627  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    37. A. Yu. Popov, “Explicit solution for the incommensurable frequency oscillation control problem under limited resource control”, Autom. Remote Control, 69:4 (2008), 597–608  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    38. Alexander A. Sherstov, “The Pattern Matrix Method”, SIAM J. Comput, 40:6 (2011), 1969  crossref
    39. A.A.. Sherstov, “The Intersection of Two Halfspaces Has High Threshold Degree”, SIAM J. Comput, 42:6 (2013), 2329  crossref
    40. F. S. Stonyakin, “Anti-compacts and their applications to analogs of Lyapunov and Lebesgue theorems in Frechét spaces”, Journal of Mathematical Sciences, 218:4 (2016), 526–548  mathnet  crossref
    41. F. S. Stonyakin, “Sequential analogues of the Lyapunov and Krein–Milman theorems in Fréchet spaces”, Journal of Mathematical Sciences, 225:2 (2017), 322–344  mathnet  crossref
    42. Yu. V. Malykhin, K. S. Ryutin, “The Product of Octahedra is Badly Approximated in the $\ell_{2,1}$-Metric”, Math. Notes, 101:1 (2017), 94–99  mathnet  crossref  crossref  mathscinet  isi  elib
    43. N. Temirgaliev, A. Zh. Zhubanysheva, “Kompyuternyi (vychislitelnyi) poperechnik v kontekste obschei teorii vosstanovleniya”, Izv. vuzov. Matem., 2019, no. 1, 89–75  mathnet
  • Успехи математических наук Russian Mathematical Surveys
    Number of views:
    This page:1354
    Full text:488
    References:66
    First page:5

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019