Using values (not solver variables x(1)...) as constraints within a minimization problem?

Hi folks,

I've got a minimization problem in matlab where not only the variables that the solver can change should be used as constraints (e.g. Microsoft Excel Solver: There you can set variables that should be changed (like Matlab), their boundaries (like Matlab) and you can set additional values as constraints like some value (not one of the variables that can be changed by the solver) shouldn't be greater than e.g. 5 [ max(value_vector)<=5 ]).

To explain it more precisely:

Do you know the Toyota Prius? It has a specially designed gearbox consisting out of planatary gear sets. One planatary gearset carry the internal combustion engine (ICE) and the 2 motors/generators (MG1 and MG2).

Prius Gearbox

On this website you can see where they are located on that gear set. Now I've got as input values the current power output of the ICE and the traction force on the wheels at some timesteps (this data is stored in a lookup table which is trigger by the simulink clock). With thoose inputs I can calculate the torques of both electric motors at every timestep (in the simulink modell). Now I'd like to minimize the size of both elektric motors (power-wise) by varrying the gear ratio x(1) of the planatary gearset (actually there is another one betweeen the ring gear, yellow on the wibsite, and MG2 which gives me a second variable x(2) ). As additional constraints I would like: none of both eletric motors should have a greater torque than a specific value during the simulation in simulink. My simulink modell includes coefficients also, that means there isn't a arithmetic solution.

Mathematically it's something like this: I would like to minimize

f = x(1) * x(2) + 5

I've got a simulink modell with a Lookup table. This table provides the input value "in_val" for the modell over time. This input value is multiplied with the two variables of the solver x(1) and x(2). The result of this multiplication is saved to the workspace as vector "out_vec" (e.g.

x(1) = 0.5
x(2) = 3
in_val = [2,4,1]
out_val = x(1) * x(2) * in_val
=> out_vec = [3,6,1.5]

). The values in "out_vec" should be less than 5

max(out_vec) <= 5

Otherwise this iteration step does not meet my needs. Is there any way to achieve this with matlab?

Currently I am using the fmincon (optimization toolbox).

I really need help on this one!

Thanks in advance for your help!

Chris

Answer 1

[Matt J]

Your problem is over-parametrized. Both your objective function and the nonlinear constraints depend on x(1) and x(2) through the product z=x(1)*x(2). Due to your bounds 0<=x(i)<=8, the product z can range between 0 and 64, and only one parameter is needed for this. By just setting x(2)=8 and treating x(1) as the only unknown, you can cover this range. The problem then reduces to the 1D optimization

    min x1

    subject to

      (8*in_vec)*x1<=5

       0<=x1<=8

If the in_vec(i) are all positive, the solution is x1=0. Otherwise, the constraint (8*in_vec)*x1<=5 is equivalent to some 1D interval [a,b] which can easily be found by simple arithmetic, and the solution is x1=max(a,0).

[Christoph]

I think I haven't explained it well enough or I simplyfied my example too much. Do you know the Toyota Prius? It has a specially designed gearbox consisting out of planatary gear sets. One planatary gearset carry the internal combustion engine (ICE) and the 2 motors/generators (MG1 and MG2).

Prius Gearbox

On this website you can see where they are located on that gear set. Now I've got as input values the current power output of the ICE and the traction force on the wheels at some timesteps. With thoose inputs I can calculate the torques of both electric motors at every timestep (in the simulink modell). Back to my example: Now I'd like to minimize the size of both elektric motors (power-wise) by varrying the gear ratio x(1) of the planatary gearset. As additional constraints I would like: none of both eletric motors should have a greater torque than a specific value during the simulation in simulink. My simulink modell includes coefficients also, that means there isn't a arithmetic solution.

I hope this describes my problem better

[Matt J]

Describing the physics of your problem only complicates the description of problem for people who aren't physicists :-( If the mathematical structure of the problem is different from what you posted, you should update your post with a more accurate mathematical description.

In particular, it might be worth describing the actual Simulink operations more, if your posted example is just a simplification. So far it looks like all you are doing with Simulink is multiplying numbers together, which is major overkill. Multiplying numbers together is the kind of thing that can be done with matrix arithmetic operations, A.*B

Answer 2

[Sean de Wolski]

Using A and b equal to the following should enforce your constraint.

I.e. max(out_vec) <= 5

A = eye(2)
b = 5+zeros(2,1)

This makes it so x1 and x2 cannot be greater than 5.

Actually, I suppose you could just use bound constraints as well.

lb = [0 0]
ub = [5 5]

http://www.mathworks.com/help/releases/R2014a/optim/ug/fmincon.html

[Andy L]

He could use optimtool to find an appropriate solver that meets his needs, which he can then program into a script.

[Christoph]

Thanks for your answers, but I guess you've got me wrong.

I have set lb and ub already for my variables x(1) and x(2), let's asume

lb = [0 0]
ub = [8 8]

Now fmincon choose 2 variables between this borders e.g.

x(1) = 0.5
x(2) = 3

After that my simulink modell is started and the values for my output vector "out_vec" are calculated. So I get a different value at each time step because of my 1D-input table "in_val".

in_val = [1,5,3]

That means at the first time step

in_val = 1

at the second

in_val = 5

and at the third time step

in_val = 3

Simulink now multiplies both input variables x(1) and x(2) with the in_val

out_vec = in_val * x(1) * x(2)

In this case my output vector for all 3 time steps would look like this

out_vec = [1.5 , 7.5 , 4.5]

The iteration step isn't succesful if one of these output values is greater than 5.

max(out_vec) <= 5

In this example the biggest value of "out_vec" is 7.5 which is greater than 5, so this iteration step does not fufill my goals.

I hope you can help me with this.

[Andy L]

Do you then need a way to indicate that the iteration step has failed, and so remove that set from your ideal input vector?

[Christoph]

Yes, a way to indicate that the iteration step has failed (because at least one of the "out_val" was greater than 5), so that the solver won't consider the solution of fmin at this step (even if it has been the best solution so far)

[Andy L]

If this is all you are trying to do a simple matrix that applies a 1 for pass and a 0 for fail could be used. You can then use this to determine the undesired values and remove them. Although I feel this is more of a workaround than a proper solution to your problem.

[Sean de Wolski]

If you are using the 'interior-point' algorithm with fmincon, you could try having your objective function return a nan if one of the values is greater than 5. This will force fmincon to take a step back to try to avoid the nan.

[Matt J]

you could try having your objective function return a nan if one of the values is greater than 5. This will force fmincon to take a step back to try to avoid the nan.

I don't know if that's a good idea. That would make the domain of the objective function a closed set and therefore non-differentiable. Differentiable objective functions and constraints must have open domains.