Biconvex optimization

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 $B\subset X\times Y$ is called a biconvex set on $X\times Y$ if for every fixed $y\in Y$, $B_{y}=\{x\in X:(x,y)\in B\}$ is a convex set in $X$ and for every fixed $x\in X$, $B_{x}=\{y\in Y:(x,y)\in B\}$ is a convex set in $Y$.

A function $f(x,y):B\to {\mathbb {R}}$ is called a biconvex function if fixing $x$, $f_{x}(y)=f(x,y)$ is convex over $Y$ and fixing $y$, $f_{y}(x)=f(x,y)$ is convex over $X$.

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

References

1. 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.