Biconvex optimization is a generalization of

convex optimization Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization probl ...

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 \subset X\times Y

is called a biconvex set on

X\times Y

if for every fixed

y\in Y

B_y = \

is a convex set in

X

and for every fixed

x\in X

B_x = \

is a convex set in

Y

. A function

f(x, y): B \to \mathbb

is called a biconvex function if fixing

x

f_x(y) = f(x, y)

is convex over

Y

and fixing

y

f_y(x) = f(x, y)

is convex over

X

. A common practice for solving a biconvex problem (which does not guarantee global optimality of the solution) is alternatively updating

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,\ldots,x_K) \to \mathbb

is block multi-convex iff it is convex with respect to each of the individual arguments while holding all others fixed.

References

Convex optimization Generalized convexity {{mathapplied-stub