Abstract
We define convexity canonically in the setting of monoids. We show that many classical results from convex analysis hold for functions defined on such groups and semigroups, rather than only vector spaces. Some examples and counter-examples are also discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atkinson, D.S., Vaidya, P.M.: A cutting plane algorithm for convex programming that uses analytic centers. Math. Progr. 69, 1, Ser. B, 1–43, Nondifferentiable and large-scale optimization (Geneva, 1992) (1995)
Blair, C.E., Jeroslow, R.G.: The value function of an integer program. Math. Progr. 23(3), 237–273 (1982)
Boland, N.L., Eberhard, A.C., Engineer, F.G., Fischetti, M., Savelsbergh, M.W.P., Tsoukalas, A.: Boosting the feasibility pump. Math. Progr. Comput. 6(3), 255–279 (2014)
Borwein, J.M.: Continuity and differentiability properties of convex operators. Proc. Lond. Math. Soc. (3) 44(3), 420–444 (1982)
Borwein, J.M., Giladi, O.: Some remarks on convex analysis in topological groups. To appear in J. Convex. Anal. (2015) Preprint available at https://www.carma.newcastle.edu.au/jon/LCGroups
Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization, 2nd ed. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 3, Springer, New York (2006) (Theory and examples)
Borwein, J.M., Penot, J.P., Théra, M.: Conjugate convex operators. J. Math. Anal. Appl. 102(2), 399–414 (1984)
Borwein, J.M., Théra, M.: Sandwich theorems for semicontinuous operators. Canad. Math. Bull. 35(4), 463–474 (1992). (English, with English and French summaries)
Borwein, J.M., Vanderwerff, J.D.: Convex functions: constructions, characterizations and counterexamples. Encyclopedia of mathematics and its applications, vol. 109. Cambridge University Press, Cambridge (2010)
Boyd, S., Vandenberghe, L.: Localization and Cutting-plane Methods. Available at http://web.stanford.edu/class/ee392o/localization-methods
Burkard, R.E., Hamacher, H., Tind, J.: On abstract duality in mathematical programming. Z. Oper. Res. Ser. A-B 26(7), A197–A209 (1982). (English, with German summary)
Fischetti, M., Glover, F., Lodi, A.: The feasibility pump. Math. Progr. 104((1, Ser. A)), 91–104 (2005)
Fuchs, L.: Infinite Abelian Groups. Vol. I, Pure and Applied Mathematics, vol. 36, Academic Press, New York (1970)
Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bull. Am. Math. Soc. 64, 275–278 (1958)
Gomory, R.E., Baumol, W.J.: Integer programming and pricing. Econometrica 28, 521–550 (1960). MR0115822 (22 #6620)
Hamel, A.: Variational Principles on Metric and Uniform Spaces, 2005. Habilitation Thesis. Available at http://sundoc.bibliothek.uni-halle.de/habil-online/05/05H167/index.htm
Holmes, R.B.: Geometric Functional Analysis and Its Applications. Springer, New York (1975). (Graduate Texts in Mathematics, No. 24)
Juutinen, P., Lu, G., Manfredi, J.J., Stroffolini, B.: Convex functions on Carnot groups. Rev. Mat. Iberoam. 23(1), 191–200 (2007)
Kaufman, R.: Interpolation of additive functionals. Studia Math. 27, 269–272 (1966)
Kelley Jr., J.E.: The cutting-plane method for solving convex programs. J. Soc. Indust. Appl. Math. 8, 703–712 (1960)
Kemprasit, Y., Triphop, N., Wasanawichit, A.: Some divisible matrix groups. Ital. J. Pure Appl. Math. 28, 225–230 (2011)
Lawson, J.: Clifford algebras, Möbius transformations, Vahlen matrices, and \(B\)-loops. Comment. Math. Univ. Carolin. 51(2), 319–331 (2010)
Letchford, A.N., Lodi, A.: Primal cutting plane algorithms revisited. Math. Methods Oper. Res. 56(1), 67–81 (2002). (Special issue on combinatorial and integer programming)
Lu, G., Manfredi, J.J., Stroffolini, B.: Convex functions on the Heisenberg group. Calc. Var. Partial Differ. Equ. 19(1), 1–22 (2004)
Matoušek, J.: Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212. Springer, New York (2002)
Mazur, S., Orlicz, W.: Sur les espaces métriques linéaires. II. Studia Math 13, 137–179 (1953). (French)
Murota, K.: Discrete Convex Analysis. Society for Industrial and Applied Mathematics (SIAM), SIAM Monographs on Discrete Mathematics and Applications, Philadelphia (2003)
Poncet, P.: Convexities on ordered structures have their Krein–Milman theorem. J. Convex Anal. 21(1), 89–120 (2014)
Robinson, D.J.S.: A course in the theory of groups, 2nd ed., Graduate Texts in Mathematics, vol. 80, Springer, New York (1996)
Rockafellar, R.T.: Convex Analysis. Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Reprint of the 1970 original; Princeton Paperbacks
Rosendal, C.: Automatic continuity of group homomorphisms. Bull. Symb. Logic 15(2), 184–214 (2009)
Tind, J., Wolsey, L.A.: An elementary survey of general duality theory in mathematical programming. Math. Progr. 21(3), 241–261 (1981)
van de Vel, M.L.J.: Theory of Convex Structures. North-Holland Mathematical Library, vol. 50, North-Holland Publishing Co., Amsterdam (1993)
Williams, H.P.: Integer programming and pricing revisited. IMA J. Math. Appl. Bus. Indust. 8(3), 203–213 (1997). (Duality in practice)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing Co., Inc., River Edge (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is dedicated to R. Tyrell Rockafellar on the occasion of his eightieth birthday.
This work was funded in part by the Australian Research Council.
Rights and permissions
About this article
Cite this article
Borwein, J.M., Giladi, O. Convex analysis in groups and semigroups: a sampler. Math. Program. 168, 11–53 (2018). https://doi.org/10.1007/s10107-016-1010-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-016-1010-x