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.

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

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

A common practice for solving a biconvex problem (which does not guarantee global optimality of the solution) is alternatively updating ${\displaystyle x,y}$ 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 ${\displaystyle f(x_{1},\ldots ,x_{K})\to \mathbb {R} }$ is block multi-convex iff it is convex with respect to each of the individual arguments while holding all others fixed.