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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.

Matt J
on 25 Feb 2018

Edited: Matt J
on 2 Mar 2018

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.

Matt J
on 3 Mar 2018

The modified objective function is linear least squares in [u,v] so lsqlin still applies.

Matt J
on 3 Mar 2018

It just requires a different choice of input matrices C,d. For example, the terms

( norm(u)^2 + norm(v)^2)

is the same as

norm( C*[u;v]-d )^2

where C=speye(2*N) and d=zeros(2*N,1).

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