Solving 3 equations with 3 unknowns gives Warning: Explicit solution could not be found > In solve at 179

I am very new to Matlab and I wrote this code to find the solutions for x, y and l for three equations. Here is the code I used;

syms x positive
syms y positive
syms m n e f l b c d

S = solve((y/((x+y)^2))*(m+e+l*d*m-l*d*n-((2*l*d+1)*((b*x)/(x+y)))+((2*l*d*b*y)/(x+y)))-l-c == 0, 
(x/((x+y)^2))*(n+f+l*d*n-l*d*m-((2*l*d+1)*((b*y)/(x+y)))+((2*l*d*b*x)/(x+y)))-l-c == 0, -x+(d*((m*x/(x+y))-(b/2)*((x/(x+y))^2)+(n*y/(x+y))-(b/2)*((y/(x+y))^2))) == 0, x, y, l )

After about 10 minutes I get the error message;

Warning: Explicit solution could not be found. 
> In solve at 179

I have googled the problem and have seen other posts pointing to the use of numerical methods, but have no idea how i would write the code.

I would greatly appreciate any help on this.

Thanks, Kai

Answer 1

[Alan Weiss]

The simplest thing to do is probably not to use symbolic variables, but to use an Optimization Toolbox solver such as fmincon. In brief, you put your variables in one vector, say x, so that x(1) = your x, x(2) = your y. Then take your objective function (the thing you are trying to maximize) as fun(x), where you write a small program to return fun(x). fmincon calculates the Lagrange multipliers automatically. You can put in your positivity constraints and other constraints easily. See this basic example.

One other thing: fmincon finds a minimum. You want a maximum. So put in -fun as your function, and minimize that.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

[Kai]

Hi Alan,

I am trying to use the Optimization Toolbox as you suggested, but my coding ability is preventing me from progressing.

I really have never used matlab before. Please could you help me set up the optimization problem in the toolbox?

Do I need to use syms at the start to define my variables and then define my objective function and constraint in two separate editors? And then use the toolbox?

Best wishes, Kai

Answer 2

[Walter Roberson]

It appears to me that a solution does exist but that it is so long that MATLAB cannot calculate it.

You are working with 4th-order equations (quartics), and symbolic solutions to quartics are seldom profitable.

What were you planning to do with the x, y, z once you had obtained them symbolically ?

[Kai]

I am solving a maximization problem subject to the restriction that x,y <= d*f(x,y) The three equations I have are the first order conditions obtained from differentiating the Lagrange multiplier with respect to x, y, and l(lambda).

One of my assumptions is that lambda>0. I need to find an expression for lambda in terms of d and this will enable me to find an expression for d in terms of a number of other parameters.

If I can get this, then I want to see how varying d changes the ratio x*/y* (the derivative by hand is far to cumbersome).

This is a model of a sports league and I didn't want to go into to much detail. Then again, my response may be very vague and if it is please let me know and I can give some background of the model.

Thanks, Kai

[Walter Roberson]

It appears to me that it is possible to get lambda in terms of d, and free of x and y, but that the resulting expression would be several tens of megabytes long at least and would not be humanly comprehensible. It would probably also be fairly sensitive to round-off errors and would probably be numerically sensitive on the inputs. This is a case where making a numeric approximation for specific inputs would probably make more sense. I would even suggest you might well be better using monte carlo methods.

[Star Strider]

When I tried solving for either ‘l’ alone or ‘[l, x, y]’ I got empty matrices ‘[]’ after a few minutes as the result. (This in 2013b.)

There may not be a symbolic solution. The numeric optimisation routines might be your best option.

[Kai]

So for numerical optimisation, do I give the parameters some values and then solve for x, y, and lambda? So then it becomes like a system of three equations with three unknowns, but not with respect to the parameters anymore.

Then I can see how the solution changes with different values of delta.

I've been trying to use fsolve for a couple of hours and I am getting nowhere. Could anyone help me with the code I for this?

[Star Strider]

Do you have the Optimization Toolbox? Alan Weiss has posted some ideas. Otherwise, consider the optimisation functions available as MATLAB core functions.

I suggest that first, you see the documentation for the solver you want to use to be sure your objective function is in a form the solver needs. Then with the Symbolic Math Toolbox, use the ‘subs’ function to convert your parameters to a subscripted parameter vector, and use ‘matlabFunction’ to convert your symbolic equations into code the solver can use.

Note that in Alan’s answer, he mentions that the solvers calculate the Lagrange multipliers, so you will have to start with your original equations, rather than the ones you posted here.

[Kai]

Yes I have the Optimization Toolbox. My original equation is below:

And my restriction is that: