MATLAB: Syntax for solving linear equations

linear

I have found out that I have phased my problem less complicated then I think it is, so I'll try again. I have a set of data points on my y-axis which has the expression: a-x*b-c*y

(a,b,c are constants, x and y are the variables to be optimized)

In an ideal case of x and y, this expression equals a straight line

a-x*b-c*y=d*z+e

(d and e are constant),

I want x and y to fit this straight line the best, so I want to optimize:

abs( a-x*b-c*y- (d*z+e) ) = 0 with the conditions (x,y >= 0 and x,y =< 1)

Best Answer

Hmm. I'm not sure what the other solvers are seeing in your question that I don't. Here is your question:

a-x*b-c*y (where a,b and c are constants).
Is minimized under the conditions that: x,y >= 0 and x,y =< 1

so, for given KNOWN constants, you want to solve for x and y. This is just a basic linear programming problem.

Note that a is irrelevant, since if you find the min of -b*x-c*y, the location does not change regardless of the value of a. Just set up linprog, assuming that you have the optimization toolbox.

TRY IT!

xy = linprog([-b,-c],[],[],[],[],[0 0],[1 1]);

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MATLAB: Solve system of equations (find the constants with respect to variable)

Do it using pencil and paper? WTP?

syms a b c d x y z

First, solve for b, and substitute.

b = -a*x

Then solve for d, using your second equation.

d = a*y - a*x - c*y
(a*z-a*x)- 2*((c*z)+ a*y - a*x - c*y)=0

Again, substitute into the other equations. This now reduces to two equations

a*x - 2*a*y + a*z + 2*c*y - 2*c*z = 0
a*(a*y - a*x - c*y) - a*x*c = 0

In the latter equation, we can see that EITHER a == 0, in which case we have b = 0, or a is non-zero. in the latter case, as long as a is non-zero...

a*x - 2*a*y + a*z + 2*c*y - 2*c*z = 0
a*y - a*x - c*y - x*c = 0

These form a homogeneous system. And as we decided above, we have a non-zero. In fact though, we can choose any value for a since the system is linear and homogeneous in the unknowns {a,c}.

Therefore, we may arbitrarily choose a == 1.

x - 2*y + z + 2*c*y - 2*c*z = 0
y - x - c*y - x*c = 0

As you see, that results in an inconsistency, since that implies two independent values for c.

c = (y-x)/(y+x)
c = (x - 2*y + z)/(2*(z-y))

So as long as a is non-zero, AND (x-y) ~= 0, AND (z-y) ~= 0, then we have a problem, UNLESS it is also true that

(y-x)/(y+x) = (x - 2*y + z)/(2*(z-y))

So unless this last relation holds for x,y,z, there is no non-trivial solution.

MATLAB: How to minimize a nonlinear function f(x,y,z) over y and z subject to the constraint that y<=x ? Is it possible to get a minimum that depends on x

xyz0 = sort(rand(1,3),'descend');     %starting point
A = [-1 1 0]; b = 0;
fmincon( @(xyz) f(xyz(1), xyz(2), xyz(3)), xyz0, A, b )

Best Answer

Related Solutions

MATLAB: Solve system of equations (find the constants with respect to variable)

MATLAB: How to minimize a nonlinear function f(x,y,z) over y and z subject to the constraint that y<=x ? Is it possible to get a minimum that depends on x

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