Optimization of function with constraints having products of variables

Hello,

I have a constrained optimization problem with an objective function involving 8 design variables (p1 to p8). I have 2 constraints that look like so:

p2 = -p4*p7 -- (i)

p3 = p6*p4 -- (ii)

I have been trying to formulate the problem using fmincon() in MATLAB but have not been successful so far in formulating the constraints as Aeq*x=beq. Could someone help me out please.

Thank you!

[John D'Errico]

Of course you cannot create products of variables in a set of LINEAR constraints. While you could create nonlinear equality constraints that do this, it will simply make the problem more complex for no good reason.

Jeff gave the correct solution. There is no need to have p2 and p3 as unknowns in the first place. They are directly computed from the other variables.

Answer 1

[Jeff Miller]

From your two constraints, it looks like p2 and p3 are completely determined by the other variables. If so, just optimize the six variables (p1, p4-p8). You can compute p2 and p3 inside the objective function, so you don't need the optimizer to know about them at all.

[shaunaksinha]

Hi Jeff, Thank you for your suggestion. I am not sure I understand how I could do this. I have a function f which looks like this:

f = p1*sin(Z)^2 + p2*sin(Z) + p3*Z_dot*sin(Z) + p4*Z_ddot*sin(Z) + p5*sin(Z)^2*cos(Z) + p6*(sin(Z)*Z_ddot - Z_dot^2*cos(Z))+p7*Z_dot*cos(Z)+p8*u*sin(Z)^2 + C

where Z and its derivatives (Z_dot, Z_ddot) are known to me at different time instants (t1 to tN). C is a term not involving any design variables but is known at each time instant. Also u is known at each instant. p1 - p8 are themselves products of certain other variables (lets call them level 0 variables). Evaluating all the p variables will help me identify the level 0 variables (which is the end goal). Now my optimization problem is

min [\summation (F_ti)]

where F_ti = f(t_i)^2

What is the best way to solve this problem? You suggested that I optimize only for (p1, p4-p8) and compute p2 and p3 inside the objective function. How can I do that since I'm passing p as a vector to fmincon as shown below?

opt_f = @(p)opt_func_sync_gen(p, y_est, y_dot_est, y_ddot_est, y_tdot_est);
options = optimoptions('fmincon', 'MaxIterations', 20, 'UseParallel' ,true);
Y = fmincon(opt_f, w0, options);

[Torsten]

By renaming p4 as p2, p5 as p3, p6 as p4, p7 as p5 and p8 as p6.

Then p2 in your objective function can be replaced by p2*p5 and p3 by p4*p2.

Best wishes

Torsten.

[Jeff Miller]

Shaunak, I don't understand everything you said, but a few comments:

(1) I agree with Torsten's suggestion; this seems like one way to accomplish your goal.

(2) Just to clarify, w0 would have only 6 elements, not 8 (maybe you already understood this). So, fmincon would be adjusting 6 "free" parameters, and you would compute the p1-p8 needed for your equation from those 6.

As a minimal example, you might have a system that involves 3 parameters, p1, p2, and p3, where p3=p1+p2. In that case you would only let fmincon adjust 2 parameters (most naturally p1 and p2), and then you would compute p3 within your objective function. (Or, as Torsten suggests, just replace p3 with (p1+p2) everywhere in the objective function.)

(3) I wonder why you aren't estimating your level 0 variables directly? From your comment, it sounds like you are computing p1-p8 from these level 0 variables. So, why not let fmincon optimize the level 0 variables directly? Each time fmincon gives a set of level 0 variables to your objective function, you would first compute p1-p8 and then use those to compute to-be-minimized error score. Maybe this cannot be done in your situation, but it seems like the most direct approach if it can be done.

Bottom line: Try to rewrite your objective function to take as few parameters as possible, and just let fmincon adjust those. Even if there are more "true parameters" in the real system, fmincon will do a better job, and faster, if you ask it to adjust the smallest possible number of parameters.

[shaunaksinha]

Thank you Torsten and Jeff! I apologize for the delay in getting back.

I tried Torsten's method and used it with fminunc. But the solution I'm getting is not what I expect (I know the right answer).

Now I plan to use fmincon with a constraint (p_i>0) since I figured that my optimal solution must satisfy this constraint.

As to Jeff's question on why I'm using the p_i variables instead of the level 0 variables, it's because most of the level 0 variables appear as products. For example p1 = E*b1. And I thought that using the p_i variables was a cleaner way to code the optimization function. However, I'm now going to code the function using the level 0 variables themselves since you say that fmincon() might give better results. I will get back as soon as I have tried these steps. Thank you!