Disciplined Multi-Convex Programming

Working draft, September 2016. Shorter version appeared in Proceedings 29th Chinese Control and Decision Conference (CCDC), pages 895–900, May 2017.

• Working draft

• CCDC paper

• DMCP Python package

• MultiConvex Julia package

A multi-convex optimization problem is one in which the variables can be partitioned into sets over which the problem is convex when the other variables are fixed. Multi-convex problems are generally solved approximately using variations on alternating or cyclic minimization. Multi-convex problems arise in many applications, such as nonnegative matrix factorization, generalized low rank models, and structured control synthesis, to name just a few. In most applications to date the multi-convexity is simple to verify by hand. In this paper we study the automatic detection and verification of multi-convexity using the ideas of disciplined convex programming. We describe an implementation of our proposed method that detects and verifies multi-convexity, and then invokes one of the general solution methods.