Abstract
An attempt is made to justify results from Convex Analysis by means of one method. Duality results, such as the Fenchel-Moreau theorem for convex functions, and formulas of convex calculus, such as the Moreau-Rockafellar formula for the subgradient of the sum of sublinear functions, are considered. All duality operators, all duality theorems, all standard binary operations, and all formulas of convex calculus are enumerated. The method consists of three automatic steps: first translation from the given setting to that of convex cones, then application of the standard operations and facts (the calculi) for convex cones, finally translation back to the original setting. The advantage is that the calculi are much simpler for convex cones than for other types of convex objects, such as convex sets, convex functions and sublinear functions. Exclusion of improper convex objects is not necessary, and recession directions are allowed as points of convex objects. The method can also be applied beyond the enumeration of the calculi.
Similar content being viewed by others
References
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Heidelberg (1998)
van den Heuvel, W., Wagelmans, A.P.M.: Worst case analysis for a general class of on-line lot-sizing heuristics. Oper. Res. (2009, accepted)
Bourbaki, N.: Elements of Mathematics. Topological Vector Spaces. Springer, Berlin (2003). Chaps. 1–5
Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Springer, Heidelberg (1993). Two volumes
Borwein, J., Lewis, A.S.: Convex Analysis and Nonlinear Optimization. Springer, New York (2000)
Minkowski, H.: Theorie der Konvexen Körper, Insbesondere Begründung Ihres Ober Flachenbegriffs. Gesammelte Abhandlungen, vol. II. Teubner, Leipzig (1911)
Fenchel, W.: On conjugate convex functions. Can. J. Math. 1, 73–77 (1949)
Fenchel, W.: Convex cones, sets, and functions. Mimeographed Notes, Princeton Univ. (1951)
Hörmander, L.: Sur la fonction d’appui des ensembles convexes dans un espace localement convexe. Ark. Mat. 3(12), 181–186 (1954)
Moreau, J.-J.: Fonctionelles convexes. Séminaire “Équations aux dérivées partielles”. Collège de France, Paris (1966)
Rockafellar, R.T.: Conjugate Duality and Optimization. Regional Conference Series in Applied Mathematics, vol. 16. SIAM, Philadelphia (1974)
Kutateladze, S.S., Rubinov, A.M.: Minkowski Duality and its Applications. Nauka, Moscow (1976)
Tikhomirov, V.M.: Fundamental Principles of the Theory of Extremal Problems. Wiley, New York (1986)
Tikhomirov, V.M.: Convex analysis. In: Gamkrelidze, R.V. (ed.) Analysis II. Encyclopaedia of Mathematical Sciences, vol. 14. Springer, Heidelberg (1980)
Magaril-Il’yaev, G.G., Tikhomirov, V.M.: Convex Analysis, Theory and Applications. Translations of Mathematical Monographs, vol. 222. AMS, Providence (2003)
Brinkhuis, J., Tikhomirov, V.M.: Optimization: Insights and Applications. Princeton University Press, Princeton (2005)
Steinitz, E.: Bedingt konvergente Reihen und konvexe Systeme, I, II, III. J. Reine Angew. Math., 143, 144–146, 128–175; 1–40; 1–52 (1913–1916)
Gauss, C.F.: Disquisitiones Generales circa Superficies Curvas (General Investigations of Curved Surfaces) of 1827 and 1825. Translated with Notes and a Bibliography by James Caddall Morehead and Adam Miller Hiltebeitel, The Princeton University Library (1902)
Hiriart-Urruty, J.-P., Lemaréchal, C.: Fundamentals of Convex Analysis. Grundlehren Text Editions. Springer, Heidelberg (2004)
Nesterov, Yu.E., Nemirovskii, A.S.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM Studies in Applied Mathematics, vol. 13. SIAM, Philadelphia (1994)
Barvinok, A.: A Course in Convexity. Graduate Studies in Mathematics, vol. 54. Am. Math. Soc., Providence (2002)
Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New York (2006)
Ben Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. SIAM, Philadelphia (2001)
Bertsekas, D., Nedić, A., Ozdaglar, A.E.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2003)
Gruber, P.M.: Convex and Discrete Geometry. Grundlehren der Mathematischen Wissenschaften, vol. 336. Springer, Heidelberg (2007)
Hörmander, L.: Notions of Convexity. Modern Birkhäuser Classics. Birkhäuser, Boston (2007)
Nesterov, Yu.E.: Introductory Lectures on Convex Optimization: A Basic Course. Kluwer Academic, Dordrecht (2004)
Grant, M., Boyd, S., Ye, Y.: Disciplined convex programming. In: Liberti, L., Macalun, M. (eds.) Global Optimization: From Theory to Implementation, pp. 155–210. Springer, Berlin (2006)
Brinkhuis, J., Tikhomirov, V.M.: Duality and calculus of convex objects. Sb. Math. 198(2), 171–206 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by B.T. Polyak.
Rights and permissions
About this article
Cite this article
Brinkhuis, J. Convex Duality and Calculus: Reduction to Cones. J Optim Theory Appl 143, 439–453 (2009). https://doi.org/10.1007/s10957-009-9574-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-009-9574-8