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Biconvex sets and optimization with biconvex functions: a survey and extensions

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  • Jochen Gorski
  • Frank Pfeuffer
  • Kathrin Klamroth

Abstract

The problem of optimizing a biconvex function over a given (bi)convex or compact set frequently occurs in theory as well as in industrial applications, for example, in the field of multifacility location or medical image registration. Thereby, a function $$f:X\times Y\to{\mathbb{R}}$$ is called biconvex, if f(x,y) is convex in y for fixed x∈X, and f(x,y) is convex in x for fixed y∈Y. This paper presents a survey of existing results concerning the theory of biconvex sets and biconvex functions and gives some extensions. In particular, we focus on biconvex minimization problems and survey methods and algorithms for the constrained as well as for the unconstrained case. Furthermore, we state new theoretical results for the maximum of a biconvex function over biconvex sets. Copyright Springer-Verlag 2007

Suggested Citation

  • Jochen Gorski & Frank Pfeuffer & Kathrin Klamroth, 2007. "Biconvex sets and optimization with biconvex functions: a survey and extensions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(3), pages 373-407, December.
    • Handle: RePEc:spr:mathme:v:66:y:2007:i:3:p:373-407
    DOI: 10.1007/s00186-007-0161-1
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    References listed on IDEAS

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    1. James E. Falk & Richard M. Soland, 1969. "An Algorithm for Separable Nonconvex Programming Problems," Management Science, INFORMS, vol. 15(9), pages 550-569, May.
    2. Simin Huang & Rajan Batta & Kathrin Klamroth & Rakesh Nagi, 2005. "The K-Connection Location Problem in a Plane," Annals of Operations Research, Springer, vol. 136(1), pages 193-209, April.
    3. Faiz A. Al-Khayyal & James E. Falk, 1983. "Jointly Constrained Biconvex Programming," Mathematics of Operations Research, INFORMS, vol. 8(2), pages 273-286, May.
    4. Willard I. Zangwill, 1969. "Convergence Conditions for Nonlinear Programming Algorithms," Management Science, INFORMS, vol. 16(1), pages 1-13, September.
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