A optimization problem with constraints

Hi, I have a optimization problem. I used the function x = fmincon(fff,x0,A,b) to do optimize ; I have some constraints on x.

Moreover, I also want to make a constraint on an intermediate variable z inside the function fff. For example let z < 0. z is in an intermediate variable of fff and will change with x.

I think fmincon cannot make a constraint on the intermediate variable z , how can I do this? Use other function instead of fmincon?

Answer 1

[Walter Roberson]

function minx = minimizex
  x0 = 1; %some guess
  A = []; b = [];, Aeq = []; beq = []; lb = []; ub = []; 
  minx = fmincon(@(x) x, X0, A, b, Aeq, beq, lb, ub, @lastz);
end

function [c, ceq] = lastz(X)
  timesteps = [.....] as appropriate
  boundaryconditions = [....] as appropriate
  zposition = 2;   %which of the outputs is z?
  %"x is parameter to the ode" 
  [T, Z] = ode45( @(t,y) YourFunction(t, y, x), timesteps, boundaryconditions);
  z = Z(end,zposition);
  c = z;
  ceq = [];
end

[craftsman]

Thanks a lot. I got it.

Answer 2

[Sean de Wolski]

You'll have to use a nonlinear constraint on x, i.e. it fails if z is outside of range.

http://www.mathworks.com/help/releases/R2015a/optim/ug/nonlinear-constraints-with-gradients.html

[Walter Roberson]

However, it might not always be feasible to calculate the ranges of x that are valid and implement them all as constraints. For example if one z = sin(50*x) and wanted to constrain -0.05 <= z <= 0.05 then if x is unconstrained then it would not be feasible to create the complete list of ranges of x where the condition will hold.

[craftsman]

Thank you a lot for your answer. The function fff is an ode function. X is the parameter of the ode, z is the result of the ode. My objective is to minimize x under the condition z < 0. Is this means the nonlinear constraint on x is the ode function itself? The nonlinear of x the ode result with parameter x?

[Matt J]

The function fff is an ode function. X is the parameter of the ode, z is the result of the ode

If z is the "result of the ode" that means z is a function of time t (or some other variable). Is that correct? If so, at what times t do you want z(t)<=0 to be satisfied? If z(t)<=0 over some infinite or continuous range of t, then fmincon cannot handle that. You might have to look at fseminf.

If instead you're happy to satisfy z(t(i))<=0 at some finite set of sample times t(i), then simply return z(t(i)) from your nonlcon function.

[craftsman]

Thank you a lot for your answer. Yes, z is the "result of the ode" and z is a function of time. The constraint on z is z(t(end)) <= 0. So I can simply set c = z(t(end)) in my nonlcon function? Is that correct?