First of all as @whuber mentioned many people are involved in non-convex optimisation. Having said, your definition of *"as long as the local minimum is the global minimum"* is rather... lax. See for instance the Easom function ($f(x,y) = -\cos(x)\cos(y)\exp(-(x-\pi)^2 - (y-\pi)^2)$. It has a single minimum if that is your concern but if your even remotely away from it you are... staffed. Standard gradient-based methods like [BFGS](https://en.wikipedia.org/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm) (`BFGS` in R's `optim` )and [Conjugate Gradient](https://en.wikipedia.org/wiki/Conjugate_gradient_method) (`CG` in R's `optim`) will suffer greatly. You will have to essentially make an "educated guess" about your answer (eg. [Simulated Annealing](https://en.wikipedia.org/wiki/Simulated_annealing) - `SANN` in R's `optim`), which a very computationally expensive routine. [![enter image description here][1]][1] 

In R:

    easom <- function(x){ 
      -cos(x[1]) * cos(x[2]) * exp( -(x[1] -pi)^2 - (x[2] - pi)^2)
    }
    optim(easom,par=c(0,0), method='BFGS')$par # 1.664149e-06 1.664149e-06 # Junk
    optim(easom,par=c(0,0), method='CG')$par # 0 0 # Insulting Junk
    optim(easom,par=c(0,0), method='SANN')$par # 3.382556 2.052309 # Some success!

There are other even worse surfaces to optimise against. See for example [Michalewicz's](http://www.sfu.ca/~ssurjano/michal.html) or [Schwefel's](http://www.sfu.ca/~ssurjano/schwef.html) functions where you might have multiple local minima and/or flat regions.

This flatness is a real problem. For example in generalised as well as standard linear mixed effects model as the number of estimation parameters increases, the log-likelihood function, even after profiling out the residual variance and the fixed-effects parameters can still be very flat. This will lead the model to converge on the boundary of the parameter space or simply to a suboptimal solution (this is actually one of the reason some people myself included are skeptical with the 'keep it maximal' idea for LME's). Therefore "*how much convex*" your objective function might have big impact to your model as well as your later inference.

  [1]: https://i.sstatic.net/qzQeZ.png