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I have a non-function (not in closed form) that takes in a few parameters (about 20) and returns a real value. A few of these parameters are discrete while others are continuous. Some of these parameters can only be chosen from a finite space of values.

Since I don't have the function in closed form, I cannot use any gradient based methods. However, the discrete nature and the boxed constraints on a few of those parameters restrict even the number of derivative free optimization techniques at my disposal. I am wondering what are the options in terms of optimization methods that I can use.

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    $\begingroup$ So you want to estimate the function and its extrema too? Or you have a black box operating as a function which you need to strategically query to determine the extremum? $\endgroup$ – Emre Dec 20 '14 at 20:42
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    $\begingroup$ I have a black box operating as a function and I need to strategically query it in order to determine the optimum. Of course, I am not looking for the global maximum but anything that is smarter than a grid search over a space of parameters. $\endgroup$ – Nitesh Dec 20 '14 at 22:16
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Bayesian optimization is a principled way of sequentially finding the extremum of black-box functions. What's more, there a numerous software packages that make it easy, such as BayesOpt and MOE. Another flexible Bayesian framework that you can use for optimization is Gaussian processes: Global Optimisation with Gaussian Processes

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