Parallel many dimensional optimization

Question

I am building a script that generates input data [parameters] for another program to calculate. I would like to optimize the resulting data. Previously I have been using the num

Answer 1

Parallelizing local optimizers is intrinsically limited: they start from a single initial point and try to work downhill, so later points depend on the values of previous evaluations. Nevertheless there are some avenues where a modest amount of parallelization can be added.

• As another answer points out, if you need to evaluate your derivative using a finite-difference method, preferably with an adaptive step size, this may require many function evaluations, but the derivative with respect to each variable may be independent; you could maybe get a speedup by a factor of twice the number of dimensions of your problem. If you've got more processors than you know what to do with, you can use higher-order-accurate gradient formulae that require more (parallel) evaluations.
• Some algorithms, at certain stages, use finite differences to estimate the Hessian matrix; this requires about half the square of the number of dimensions of your matrix, and all can be done in parallel.

Some algorithms may also be able to use more parallelism at a modest algorithmic cost. For example, quasi-Newton methods try to build an approximation of the Hessian matrix, often updating this by evaluating a gradient. They then take a step towards the minimum and evaluate a new gradient to update the Hessian. If you've got enough processors so that evaluating a Hessian is as fast as evaluating the function once, you could probably improve these by evaluating the Hessian at every step.

As far as implementations go, I'm afraid you're somewhat out of luck. There are a number of clever and/or well-tested implementations out there, but they're all, as far as I know, single-threaded. Your best bet is to use an algorithm that requires a gradient and compute your own in parallel. It's not that hard to write an adaptive one that runs in parallel and chooses sensible step sizes for its numerical derivatives.