Biconvex sets and optimization with biconvex functions: a survey and extensions |
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Authors: | Jochen Gorski Frank Pfeuffer Kathrin Klamroth |
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Institution: | (1) Institute for Applied Mathematics, Friedrich-Alexander-University Erlangen-Nuremberg, Erlangen, Germany |
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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 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.
J. Gorski and K. Klamroth were partially supported by a grant of the German Research Foundation (DFG). |
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Keywords: | Biconvex functions Biconvex sets Biconvex optimization Biconcave optimization Non-convex optimization Generalized convexity |
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