Biconvex optimization

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Biconvex optimization is a generalization of convex optimization where the objective function and the constraint set can be biconvex. There are methods that can find the global optimum of these problems.[1][2]

A set is called a biconvex set on if for every fixed , is a convex set in and for every fixed , is a convex set in .

A function is called a biconvex function if fixing , is convex over and fixing , is convex over .

A common practice for solving a biconvex problem (which does not guarantee global optimality of the solution) is alternatively updating by fixing one of them and solving the corresponding convex optimization problem.[1]

The generalization to functions of more than two arguments is called a block multi-convex function. A function is block multi-convex iff it is convex with respect to each of the individual arguments while holding all others fixed.[3]


  1. ^ a b Gorski, Jochen; Pfeuffer, Frank; Klamroth, Kathrin (22 June 2007). "Biconvex sets and optimization with biconvex functions: a survey and extensions" (PDF). Mathematical Methods of Operations Research. 66 (3): 373–407. doi:10.1007/s00186-007-0161-1.
  2. ^ Floudas, Christodoulos A. (2000). Deterministic global optimization : theory, methods, and applications. Dordrecht [u.a.]: Kluwer Academic Publ. ISBN 978-0-7923-6014-8.
  3. ^ Chen, Caihua (2016). ""The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent"". "Math. Prof.". 155: 57–59. doi:10.1007/s10107-014-0826-5.