Many situations give rise to sparse linear least-squares problems,
often with bounds on the variables. You can use the
to solve sparse bound-constrained problems. 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); options = optimoptions('lsqlin','Algorithm','trust-region-reflective'); [x,resnorm,residual,exitflag,output] = ... lsqlin(C,d,,,,,lb,,,options);
Optimization terminated: relative function value changing by less than sqrt(OPTIONS.FunctionTolerance), and rate of progress is slow.
The default diagonal preconditioning works fairly well:
exitflag,resnorm,output exitflag = 3 resnorm = 22.5794 output = struct with fields: 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
defined as in Box Constraints, and
You can improve (decrease) the first-order optimality measure
by using a sparse QR factorization in each iteration. To do this,
options = optimoptions(options,'SubproblemAlgorithm','factorization'); [x,resnorm,residual,exitflag,output] = ... lsqlin(C,d,,,,,lb,,,options);
Optimization terminated: first order optimality with optimality gradient norm less than OPTIONS.OptimalityTolerance.
The first-order optimality measure decreases:
exitflag,resnorm,output exitflag = 1 resnorm = 22.5794 output = struct with fields: iterations: 12 algorithm: 'trust-region-reflective' firstorderopt: 5.5907e-15 cgiterations: 0 message: 'Optimization terminated: first order optimality with optimality…'