Английская Википедия: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.

A set <math>B \subset X\times Y </math> is called a biconvex set on <math>X\times Y </math> if for every fixed <math>y\in Y </math>, <math>B_y = \{x \in X: (x,y) \in B\}</math> is a convex set in <math>X </math> and for every fixed <math>x\in X </math>, <math>B_x = \{y \in Y: (x,y) \in B\}</math> is a convex set in <math>Y </math>.

A function <math>f(x, y): B \to \mathbb{R} </math> is called a biconvex function if fixing <math>x</math>, <math> f_x(y) = f(x, y)</math> is convex over <math> Y </math> and fixing <math>y</math>, <math> f_y(x) = f(x, y)</math> is convex over <math> X </math>.

A common practice for solving a biconvex problem (which does not guarantee global optimality of the solution) is alternatively updating <math>x, y</math> 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 <math> f(x_1,\ldots,x_K) \to \mathbb{R} </math> is block multi-convex iff it is convex with respect to each of the individual arguments while holding all others fixed.

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Шаблон:Mathapplied-stub