Question about 2 output optimization

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Kokalz 2014-9-24

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评论： Kokalz 2014-10-7

Hi guys!

I'm struggling to formulate a typical optimization problem into something than can be solved by MATLAB and was hoping you could assist me with this. I do have optimization toolbox available on my machine, so can use any function offered by the toolbox.

I have a software simulating a very complex non-linear model, that takes a set of inputs and gives me 2 outputs. In this case I am running a batch of simulations, only changing 3 input parameters, so I basically have a matrix of 3 input values and a matrix of output values. My aim is to correlate 1-st output parameter to match real-life data as closely as possible, while keeping the 2nd output parameter equal to a certain value. In other words I am trying to minimize error between real-life data and simulated data (what would be a typical error function for such case?), while imposing an equality constraint on the second output. I obviously cannot write the whole simulation in terms of single f(x) function to minimize.

Any thoughts on how to write such optimization problem and what function is typically used for something like this ?

Thank you for any help!

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Answer 1

Alan Weiss 2014-9-24

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This sounds straightforward to solve in Optimization Toolbox. You have a nonlinear equality constraint (the value of the second output) and want to minimize the difference between the first output and some data that you already have (maybe the sum of squares of the differences of the components if it is multidimensional).

You can easily formulate this for fmincon. To save computation time, use the technique described here for the case when your nonlinear constraint and objective function are given by the same computation.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

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Kokalz 2014-9-24

编辑：Kokalz 2014-9-24

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Hi Alan,

It's not urgent, so any help is highly appreciated. I think I understand what you were saying though. I'll try to explain myself a little better this time.

Basically I have a large non-linear model, that takes in 2 input values and spits out 2 output values. Lets shows this as:

[in1, in2] -> {simulation model} -> [out1 out2]

I do not have access to the stuff going on inside the simulation model, only the input and output parameters. I can run say a batch of 250 simulations to improve the sensitivity study.

Now, in1 and in2 are single numbers, such as

in1 = 100, 
in2 = 200

However:

out1 = [1x1000] % vector
out2 = [1x1] % value

I also have real-life experimental data:

RLout1 = [1x1000] 
RLout2 = [1x1]

My aim is to tune the in1 and in2 parameters such that:

1) Minimise the error between out1 and RLout1, while
2) Have out2 = RLout2

So that basically makes it a a typical optimization problem, where I am minimizing an error (sum of squared errors ?) between the out1 and RLout1 vectors, while imposing the equality constraint.

I guess what you are saying is that my error is functioning as f(x) that I need to minimize. I can write that function as

min f(x) = SSE(RL_Data, predicted_Data)

But I just can't get my head around how I would be relating my simulated data and the optimization function. What would be the function linking real life data, simulated data and optimization predictions ? Do I fit a quadratic equation to my error between real life data and simulated data and then use that equation for optimization? Does this question make sense ?

Huge thank you for your help!

Alan Weiss 2014-10-6

If it is impossible for MATLAB to communicate with your simulation, then I do not see how you can profitably use MATLAB optimization functions. The functions query various values of the control variables (x) and use the resulting objective function values to guide a search for the lowest objective function consistent with constraints.

But an idea just popped into my head. I suppose that you might write a grid of control variable values, evaluate the objective and constraints on this grid, and then construct an interpolating function that fmincon could evaluate. For an example using this technique, see Pattern Search Climbs Mt. Washington. The objective function, terrainfun, is just a call to interp2. You could do the same with the nonlinear constraint.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

Kokalz 2014-10-7