2
$\begingroup$

I have a function $f(\boldsymbol{x})$ of $d>2$ variables that is piecewise convex. By that, I mean that there are regions in the domain $\boldsymbol{x}$ such that if the domain was restricted to those regions, then $f$ would appear convex. The actual function $f$ is formed by "stitching" each of these piecewise convex shapes together. In a typical example, $d=4$ and the number of piecewise regions might be $50$.

I am trying to find the global minimum of this function. Is there a good, systematic way to find the minimum? Right now, I am just trying a bunch of different initialization points, which is exceedingly slow.

  • 2
    Can you solve 50 subproblems, each with a constraint that the feasible set is restricted to where the function is convex? – 2017-01-27
  • 0
    I suppose so, but I don't know exactly where each piecewise convex region starts and ends. They are randomly spread out. I guess I was wondering whether this is a standard type of minimization problem, and whether there was a standard way of approaching these problems? – 2017-01-28
  • 0
    Local information no longer provides information about where to find the global optimum. Can you tell more about how the function $f$, like how you obtain it or what form it has? – 2017-01-28
  • 2
    Indeed, I was going to joke: what do you call a piecewise convex function? A non-convex function ;-) Seriously, unless you can say more about the regions of convexity, you're out of luck. – 2017-01-28

0 Answers 0