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

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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$f:X\times Y\to{\mathbb{R}}$$</EquationSource> </InlineEquation> 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