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.{{cite journal|last=Gorski|first=Jochen|author2=Pfeuffer, Frank |author3=Klamroth, Kathrin|author3-link= Kathrin Klamroth |title=Biconvex sets and optimization with biconvex functions: a survey and extensions|journal=Mathematical Methods of Operations Research|date=22 June 2007|volume=66|issue=3|pages=373–407|doi=10.1007/s00186-007-0161-1|s2cid=15900842 |url=http://www2.math.uni-wuppertal.de/~klamroth/publications/gopfkl07.pdf}}{{cite book|last=Floudas|first=Christodoulos A.|authorlink1=Christodoulos Floudas|title=Deterministic global optimization : theory, methods, and applications|year=2000|publisher=Kluwer Academic Publ.|location=Dordrecht [u.a.]|isbn=978-0-7923-6014-8|url=https://www.springer.com/mathematics/book/978-0-7923-6014-8}}

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

A function $f(x,\; y):\; B\; \backslash to\; \backslash mathbb\{R\}$ 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.

The generalization to functions of more than two arguments

is called a block multi-convex function.

A function

$$

f(x_1,\ldots,x_K) \to \mathbb{R}

is block multi-convex

iff it is convex with respect to each of the individual arguments

while holding all others fixed.{{cite journal

|last=Chen

|first=Caihua

|title="The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent"

|journal= Mathematical Programming

|volume=155

|pages=57–59

|doi=10.1007/s10107-014-0826-5

|year=2016

|issue=1–2

|s2cid=5646309

}}