Non Linear Least Squares with Inequality Constraints
Parse linear equations/inequalities
Join elements into a string
Least Distance Problem
Linear Least Squares, least norm solution
Linear Least Squares, least norm solution (by svd)
Linear Least Squares with Inequality constraints (LSI)
Linear Least Squares with Inequality constraints, least norm solution
Regularized Linear Least Squares
Linear Least Squares problem with inequality and equality constraints,...
Non Linear Least Squares with Inequality Constraints
Null-space basis
Total Least Squares a%*%x ~= b
Transform box-type inequalities into matrix and vector form
We solve non linear least squares problems with optional equality and/or inequality constraints. Non linear iterations are globalized with back-tracking method. Linear problems are solved by dense QR decomposition from 'LAPACK' which can limit the size of treated problems. On the other side, we avoid condition number degradation which happens in classical quadratic programming approach. Inequality constraints treatment on each non linear iteration is based on 'NNLS' method (by Lawson and Hanson). We provide an original function 'lsi_ln' for solving linear least squares problem with inequality constraints in least norm sens. Thus if Jacobian of the problem is rank deficient a solution still can be provided. However, truncation errors are probable in this case. Equality constraints are treated by using a basis of Null-space. User defined function calculating residuals must return a list having residual vector (not their squared sum) and Jacobian. If Jacobian is not in the returned list, package 'numDeriv' is used to calculated finite difference version of Jacobian. The 'NLSIC' method was fist published in Sokol et al. (2012) <doi:10.1093/bioinformatics/btr716>.