Biconvex optimization explained

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

B\subsetX x Y

is called a biconvex set on

X x Y

if for every fixed

y\inY

,

By=\{x\inX:(x,y)\inB\}

is a convex set in

X

and for every fixed

x\inX

,

Bx=\{y\inY:(x,y)\inB\}

is a convex set in

Y

.

A function

f(x,y):B\toR

is called a biconvex function if fixing

x

,

fx(y)=f(x,y)

is convex over

Y

and fixing

y

,

fy(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 argumentsis called a block multi-convex function.A function

f(x1,\ldots,xK)\toR

is block multi-convexiff it is convex with respect to each of the individual argumentswhile holding all others fixed.[3]

Notes and References

  1. Gorski. Jochen. Pfeuffer, Frank . Klamroth, Kathrin. Kathrin Klamroth . Biconvex sets and optimization with biconvex functions: a survey and extensions. Mathematical Methods of Operations Research. 22 June 2007. 66. 3. 373–407. 10.1007/s00186-007-0161-1. 15900842 .
  2. Book: Floudas, Christodoulos A.. Christodoulos Floudas. Deterministic global optimization : theory, methods, and applications. 2000. Kluwer Academic Publ.. Dordrecht [u.a.]. 978-0-7923-6014-8.
  3. Chen. Caihua. "The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent". Mathematical Programming. 155. 57–59. 10.1007/s10107-014-0826-5. 2016. 1–2. 5646309.