Many situations give rise to sparse linear least-squares problems, often with bounds on the variables. The next problem requires that the variables be nonnegative. This problem comes from fitting a function approximation to a piecewise linear spline. Specifically, particles are scattered on the unit square. The function to be approximated is evaluated at these points, and a piecewise linear spline approximation is constructed under the condition that (linear) coefficients are not negative. There are 2000 equations to fit on 400 variables:
load particle % Get C, d
lb = zeros(400,1);
[x,resnorm,residual,exitflag,output] = ...
lsqlin(C,d,[],[],[],[],lb);The default diagonal preconditioning works fairly well:
exitflag,resnorm,output
exitflag =
3
resnorm =
22.5794
output =
iterations: 10
algorithm: 'trust-region-reflective'
firstorderopt: 2.7870e-05
cgiterations: 42
message: 'Optimization terminated: relative function value changing by less…'For bound constrained problems, the first-order optimality is
the infinity norm of v.*g, where v is
defined as in Box Constraints, and g is
the gradient.
You can improve (decrease) the first-order optimality measure
by using a sparse QR factorization in each iteration. To do this,
set PrecondBandWidth to inf:
options = optimoptions('lsqlin','PrecondBandWidth',inf);
[x,resnorm,residual,exitflag,output] = ...
lsqlin(C,d,[],[],[],[],lb,[],[],options);The first-order optimality measure decreases:
exitflag,resnorm,output
exitflag =
1
resnorm =
22.5794
output =
iterations: 12
algorithm: 'trust-region-reflective'
firstorderopt: 5.5907e-15
cgiterations: 0
message: 'Optimization terminated: first order optimality with optimality…'