Solve minimax constraint problem

Hi,

I would like to solve minmax problem. like below:

So first of all I try to use two fmincon but as it is slow I noticed Matlab offer fminmax function: http://www.mathworks.co.uk/help/optim/ug/fminimax.html

So I would like to ask is it possible to give the range to 'i' in fminmax function?

With best regards,

Answer 1

[Matt J]

So I would like to ask is it possible to give the range to 'i' in fminmax function?

You will need to discretely sample t in order for fminimax to be applicable. You also need to optimize over a region where the F_i are differentiable, so if p<=1 in your p-norms, you will also have to keep Px, Qx and Rx away from zero.

[Sina]

Thanks for you answer, But my problem is very basic. fminmax function of Matlab looks like below:

x = fminimax(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

and based on my formulation above I only manage to give the range of "u" to it and I can't give the range of "w" to it.

umax<u<umin
wmax<w<wmin

u and w are x and i the the Matlab fmincon function.

[Matt J]

If U_t is really a discrete vector, then your problem isn't really a minimax in the standard sense. You can rewrite the problem as a continuous minimization problem in both w and u,

min_{w,u}(-J0(w,u))

So, it's really just a regular minimization problem in the concatenated space of vectors z=[w,u]

min_z -J0(z)

[Sina]

Thanks,

Actually you are right about overall cases but the idea of my code is to find the maximum value of the cost function when disturbance occurs in the certain range of wmin and wmax and then try to find the opimal manipulated variable which can minimize our cost function in the present of the maximum disturbance. So u and w are different values which must stay constant in each min max. u is constant in maximum part and w is constant in mimimum part. The main problem for using two fmincon is the computation time so maybe fminmax is better if I can define my problem there.