Abstract
We generalize a well known convexity property of the multiplicative potential function. We prove that, given any convex function \({g : \mathbb{R}^m \rightarrow [{0}, {\infty}]}\), the function \({({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}\), is convex if β ≥ 0 and α ≥ β 1 + ··· + β n . We also provide further generalization to functions of the form \({({\rm \bf x},{\rm \bf y}_1, . . . , {y_n})\mapsto g({\rm \bf x})^{1+\alpha}f_1({\rm \bf y}_1)^{-\beta_1} \cdot \cdot \cdot f_n({\rm \bf y}_n)^{-\beta_n} }\) with the f k concave, positively homogeneous and nonnegative on their domains.
Similar content being viewed by others
References
Avriel M., Diewert W.E., Schaible S., Zang I.: Generalized Convexity. Plenum Press, New York (1988)
Borwein J. M.: A generalization of Young’s l p inequality. Math. Inequalities Appl. 1, 131–136 (1998)
Crouzeix J.P., Ferland J.A., Schaible S.: Generalized convexity on affine subspaces with an application to potential functions. Math. Program 56, 223–232 (1992)
Hiriart-Urruty J.-B., Lemaréchal C.: Convex Analysis and Minimization Algorithms, I and II. Springer, Berlin (1993)
Imai I.: On the convexity of the multiplicative version of Karmarkar’s potential function. Math. Program 40, 29–32 (1988)
Iri, M., Imai, I.: A multiplicative penalty function method for linear programming—another ’new and fast’ algorithm. In: Proceedings of the 6th Mathematical Programming Symposium of Japan, pp. 97–120, Tokyo (1985)
Iri M., Imai I.: A multiplicative barrier function method for linear programming. Algorithmica 1, 455–482 (1986)
Konno H., Kuno T., Yajima Y.: Global minimization of a generalized convex multiplicative function. J. Glob. Optim. 4(1), 47–62 (1994)
Konno H., Fukaishi K.: A branch and bound algorithm for solving low rank linear multiplicative and fractional programming problems. J. Glob. Optim. 18(3), 283–299 (2000)
Maréchal P.: On the convexity of the multiplicative and penalty functions and related topics. Math. Program. Ser. A 89, 505–516 (2001)
Maréchal P.: On a functional operation generating convex functions. Part I: duality. J. Optim. Theory Appl. 126(1), 175–189 (2005)
Maréchal P.: On a functional operation generating convex functions. Part II: algebraic properties. J. Optim. Theory Appl. 126(2), 357–366 (2005)
Maréchal P.: On a class of convex sets and functions. Set Valued Anal. 13, 197–212 (2005)
Matsui T.: NP-hardness of linear multiplicative programming and related problems. J. Glob. Optim. 9(2), 113–119 (1996)
Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Maréchal, P. A convexity theorem for multiplicative functions. Optim Lett 6, 357–362 (2012). https://doi.org/10.1007/s11590-011-0277-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-011-0277-3