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.[1][2]

A set is called a biconvex set on if for every fixed , is a convex set in and for every fixed , is a convex set in .

A function is called a biconvex function if fixing , is convex over and fixing , is convex over .

A common practice for solving a biconvex problem (which does not guarantee global optimality of the solution) is alternatively updating 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 is block multi-convex iff it is convex with respect to each of the individual arguments while holding all others fixed.[3]

References

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