[mccourt]

First off, I am very hesitant to say anything about biconvex problems - I only see them in passing and they are definitely not in my wheelhouse. If anyone out there is an expert, or even just has a solid (basic) reference on biconvex problems, please feel free to drop some knowledge on me.

For this particular problem, which has only one input variable, yes the answer can be resolved with a good-old fashioned Plug-In-The-Answer strategy. For problems with more than one input variable, that will almost certainly not be the case.

Really, all I was trying to say there at the end is that converting the multicriteria problem to a constraint based problem has potential benefits over scalarization. Speaking only for myself, I always default to treating multicriteria problems in some sort of norm-scalarized sense: minimize ||g|| for some vector norm. I thought it was valuable to remind myself, and maybe others, that there are other ways to naturally rephrase multicriteria problems as scalar optimization problems. I'm definitely not saying anything about how easy it is to solve, as in general these constrained problems are going to be harder than the non-constrained linear (or norm) scalarization.

[LolWolf]

The usual reference is this[1] paper which outlines most well-known results on biconvexity and optimiality thereof.

[1] http://www2.math.uni-wuppertal.de/~klamroth/publications/gop...

[mccourt]

That's very helpful. Thanks a lot!

[apathy]

That makes sense, at least as long as the vector- or matrix-valued objective behaves somewhat. I guess I've been using this all along (with transformations to enforce proper behavior) for matrix and tensor completion anyways... Hmm. I just didn't implement it terribly elegantly!

Now that I think about it, all of the methods I've ever seen for matrix-valued time series fits (i.e. multiple measurements at multiple sites per time point) are Bayesian. That's about the most irreducible constrained optimization problem I can think of in this setting.

[mccourt]

Do you have a reference for fitting matrix-valued time series with nonlinear criteria? I'm familiar with the standard Box-Jenkins methods but I usually see that done with linear least-squares methods. I'd love to up my game on that front.

[apathy]

I'm buried under manuscripts right now but making a mental note to look up Mike West's notes when I get home.