# Black-box optimization with linear programming?

In my research, I do a black-box optimization based on a simulation model with nonlinear properties. The simulation model gets an operation plan for a time period and then returns a time series, which is evaluated in the fitness function of the algorithms. There are several local minima in the state space. For optimization, I use heuristic algorithms like Particle Swarm Optimization (PSO) or Simulated Annealing (SA).

For this optimization problem, I have already written several papers and each time the reviewers mention why I don't use methods like Mixed integer linear programming, dynamic programming, quadratic programming, etc.

So far, I have assumed that these mathematical methods do not make sense in my case. How do you see this?

• To my knowledge, black boxes are used when the problem is too complex to be modelled by mixed linear programming, or when the black box is part of the problem in the first place. Is this your case ? – Kuifje Nov 27 '19 at 8:50
• Can you link to the papers that you mention? – Kevin Dalmeijer Nov 27 '19 at 13:19
• Just in case there was a misunderstanding... There is another algorithm for black box optimization called the simplex method that has nothing to do with the simplex method for linear programming. Is it possible that this was what the referees were talking about? – Brian Borchers Dec 1 '19 at 1:16

My experience in this may be a bit dated (it comes from a previous millennium), but back then I recall (vaguely) using a form of response surface methodology to optimize parameters in a simulation model. The idea was to run the model with a range of parameter values and harvest observations, fit a nonlinear model statistically (with the performance measure as the dependent variable), optimize that function, and optionally simulate again using parameter values in a neighborhood of the alleged optimum, refit, reoptimize, ...

Given that the response surface is almost always nonlinear, using a MIP model strikes me as unlikely. A quadratic program (or quadratic MIP) might work. That still leaves a couple of questions. The first is whether the parameters are constrained in known ways. (MILP and MIQP generally assume there are linear constraints.) The second is whether worrying about closing an optimality gap makes sense when (a) you are dealing with a somewhat coarse approximation of the actual response function and (b) you are going to do this iteratively. In my graduate student days (at a school with strong agricultural college), we referred to this as "milking a duck".

Personally, I would use a "good" heuristic or metaheuristic. Feel free to suggest (tactfully) to reviewers what I said above about optimality gaps of very approximate response functions and iteration/repetition.

• Thank you very much for the helpful answer, it helped me a lot. – Emma Dec 3 '19 at 13:11

AFAIK, it depends on the optimization problem under study. As @Kuifje said, black boxes are used when the problem is too complex.

One of the ways to apply simulation-optimization is to use discrete event simulation to calculate the results of the complex problem and then, feeding that into the model which can be represented using the mixed-integer programming.

An interesting example can be a blood supply chain optimization under uncertainty. However, there are many articles which authors have used such a method to do that.