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clsSolve

Solves sparse or dense nonlinear least squares optimization problems with explicit handling of linear inequality and equality constraints and simple bounds on the variables. The solver is suitable for ill-conditioned nonlinear least squares problems and nonlinear systems of equations. It may also be used for linear least squares.

Seven methods are implemented:

Method Reference
Fletcher-Xu hybrid method (default)  Fletcher, Xu 1987
Al-Baali-Fletcher hybrid method Al-Baali, Fletcher 1986
Huschens TSSM method Huschens, SIAM J. Optimization. Vol 4, No 1, pp 108-129,1994
Gauss-Newton with subspace minimization
Wang, Li, Qi Structured MBFGS method
Li-Fukushima MBFGS method
Broydens method

  Main features:

  • An active-set strategy by Björkman and Holmström handles simple bounds and linear constraints
     
  • If rank problem occurs, the solver is using subspace minimization techniques. A rank tolerance parameter may be set by the user.
     
  • The equation systems are solved with sparse or dense QR-factorization, in both cases using pivoting. Using dense SVD (singular value decomposition) is an option.
     
  • If the flag Prob.LargeScale is set, clsSolve is using the sparse QR package sqr2. clsSolve then can avoid forming the m by m orthogonal Q matrix. Otherwise, if the number of residuals, m, is very large, this Q matrix occupies very much memory.
     
  • For illustrative purposes, the equation system may also be solved with the pseudoinverse (pinv) in Matlab or the built-in inverse. These methods are not for practical use.
     
  • The line search is a modified version of an algorithm by Fletcher (1987)
     
  • The hybrid methods are using BFGS safeguarded quasi-Newton updates. The initial matrix may be given by the user.
     
  • The initial feasible point is found solving a special quadratic programming problem. This problem is solved by QPOPT in Tomlab /MINOS or qpSolve in the Tomlab Base Module.
     
  • If missing, unknown gradients and Jacobians are estimated using any of the Tomlab methods.
     
  • Since no second order derivative information is used, clsSolve may not be able to determine the type of stationary point converged to.
     

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