Biconvex sets and optimization with biconvex functions: a survey and extensions
Biconvex sets and optimization with biconvex functions: a survey and extensions
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DOI：
10.1007/s00186-007-0161-1
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发表时间：
2007-12-01
影响因子：
1.2
通讯作者：
Klamroth, Kathrin
中科院分区：
文献类型：
--
作者：
Gorski, Jochen;Pfeuffer, Frank;Klamroth, Kathrin
在给定的（双）凸或紧集上优化双凸函数的问题在理论和工业应用中经常出现，例如在多设施定位或医学图像配准领域。因此，函数 f : X x Y -> R 被称为双凸，如果固定 x 是 X 的元素，f(x,y) 在 y 上是凸的，并且当固定 y 是 Y 的元素时，f(x,y) 在 x 上是凸的。本文概述了有关双凸集和双凸函数理论的现有结果，并给出了一些扩展。我们特别关注双凸最小化问题以及受约束和无约束情况的调查方法和算法。此外，我们还提出了双凸集上双凸函数最大值的新理论结果。
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 x Y -> R is called biconvex, if f(x,y) is convex in y for fixed x is an element of X, and f(x,y) is convex in x for fixed y is an element of 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.