Mathematical Methods of Operations Research

, Volume 66, Issue 3, pp 373–407 | Cite as

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

Original Article

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 xX, and f(x,y) is convex in x for fixed yY. 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.

Keywords

Biconvex functions Biconvex sets Biconvex optimization Biconcave optimization Non-convex optimization Generalized convexity 

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Jochen Gorski
    • 1
  • Frank Pfeuffer
    • 1
  • Kathrin Klamroth
    • 1
  1. 1.Institute for Applied MathematicsFriedrich-Alexander-University Erlangen-NurembergErlangenGermany

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