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.^{[1] [2]}

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

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

A common practice for solving a biconvex problem (which does not guarantee global optimality of the solution) is alternatively updating ${\displaystyle x,y}${\displaystyle x,y} by fixing one of them and solving the corresponding convex optimization problem.^[1]

The generalization to functions of more than two arguments is called a block multi-convex function. A function ${\displaystyle f(x_{1},\ldots ,x_{K})\to \mathbb {R} }${\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.^[3]

• Convex optimization
• Generalized convexity
• Applied mathematics stubs

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