MATLAB: Solve linear system of equations with constains

constaintlinear system of equationslsqlinoptimization

Hello everyone. I have a linear system of equations that make a matrix, L*x=R. x is composed on many variables, e.g. x=[x1 x2 x3 x4 … xN]. I want to solve this system of equation with constraints x1>|x2|>|x3|>|x4|…>|xN|. Can I use lsqlin(L,R) with some additional input to realize it?
Many thanks.

Best Answer

If you make the change of variables x(i)=u(i)-v(i) with linear constraints
u(i)>=0,
v(i)>=0, 
u(i)+v(i)>=u(i+1)+v(i+1)
and modify your least squares objective from norm(L(u-v)-R)^2 to
   norm(L(u-v)-R)^2 + C*( norm(u)^2 + norm(v)^2)
then for a sufficiently small choice of C>0, this should give an equivalent solution. It's not ideal, since it forces you to re-solve with multiple choices of C, but on the other hand, it allows you to pose this as a convex problem.
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