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. 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 Bsubset X imes Y} is called a biconvex set on X × Y {displaystyle X imes Y} if for every fixed y ∈ Y {displaystyle yin Y} , B y = { x ∈ X : ( x , y ) ∈ B } {displaystyle B_{y}={xin X:(x,y)in B}} is a convex set in X {displaystyle X} and for every fixed x ∈ X {displaystyle xin X} , B x = { y ∈ Y : ( x , y ) ∈ B } {displaystyle B_{x}={yin Y:(x,y)in B}} is a convex set in Y {displaystyle Y} . A function f ( x , y ) : B → R {displaystyle f(x,y):B o mathbb {R} } is called a biconvex function if fixing x {displaystyle x} , f x ( y ) = f ( x , y ) {displaystyle f_{x}(y)=f(x,y)} is convex over Y {displaystyle Y} and fixing y {displaystyle y} , f y ( x ) = f ( x , y ) {displaystyle f_{y}(x)=f(x,y)} is convex over X {displaystyle X} . A common practice for solving a biconvex problem (which does not guarantee global optimality of the solution) is alternatively updating x , y {displaystyle x,y} by fixing one of them and solving the corresponding convex optimization problem. The generalization to functions of more than two argumentsis called a block multi-convex function.A function f ( x 1 , … , x K ) → R {displaystyle f(x_{1},ldots ,x_{K}) o mathbb {R} } is block multi-convexiff it is convex with respect to each of the individual argumentswhile holding all others fixed.