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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 B ⊂ X × Y {\displaystyle B\subset X\times Y} is called a biconvex set on X × Y {\displaystyle X\times Y} if for every fixed y ∈ Y {\displaystyle y\in Y} , B y = { x ∈ X : ∈ B } {\displaystyle B_{y}=\{x\in X:\in B\}} is a convex set in X {\displaystyle X} and for every fixed x ∈ X {\displaystyle x\in X} , B x = { y ∈ Y : ∈ B } {\displaystyle B_{x}=\{y\in Y:\in B\}} is a convex set in Y {\displaystyle Y}.
A function f : B → R {\displaystyle f:B\to \mathbb {R} } is called a biconvex function if fixing x {\displaystyle x} , f x = f {\displaystyle f_{x}=f} is convex over Y {\displaystyle Y} and fixing y {\displaystyle y} , f y = f {\displaystyle f_{y}=f} is convex over X {\displaystyle X}.
A common practice for solving a biconvex problem is alternatively updating x , y {\displaystyle x,y} by fixing one of them and solving the corresponding convex optimization problem.