期刊
MATHEMATICAL METHODS OF OPERATIONS RESEARCH
卷 66, 期 3, 页码 373-407出版社
SPRINGER HEIDELBERG
DOI: 10.1007/s00186-007-0161-1
关键词
biconvex functions; biconvex sets; biconvex optimization; biconcave optimization; non-convex optimization; generalized convexity
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.
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