Biconvex optimization

Biconvex optimization is a generalization of <a href="/facts/Convex_optimization/7D6U4MTN">convex optimization</a> 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
 :
 (
 x
 ,
 y
 )
 ∈
 B
 }
 
 
 {\displaystyle B_{y}=\{x\in X:(x,y)\in B\}}
 
 is a convex set in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and for every fixed 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
, 
 
 
 
 
 B
 
 x
 
 
 =
 {
 y
 ∈
 Y
 :
 (
 x
 ,
 y
 )
 ∈
 B
 }
 
 
 {\displaystyle B_{x}=\{y\in 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\to \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 arguments
is called a block multi-convex function.
A function

f
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            K
          
        
        )
        →
        
          R
        
      
    
    {\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.

Biconvex optimization open-in-new

Biconvex optimization