Class LevenbergMarquardtOptimizer

All Implemented Interfaces:
BaseMultivariateVectorOptimizer<DifferentiableMultivariateVectorFunction>, BaseOptimizer<PointVectorValuePair>, DifferentiableMultivariateVectorOptimizer

@Deprecated public class LevenbergMarquardtOptimizer extends AbstractLeastSquaresOptimizer
Deprecated.
As of 3.1 (to be removed in 4.0).
This class solves a least squares problem using the Levenberg-Marquardt algorithm.

This implementation should work even for over-determined systems (i.e. systems having more point than equations). Over-determined systems are solved by ignoring the point which have the smallest impact according to their jacobian column norm. Only the rank of the matrix and some loop bounds are changed to implement this.

The resolution engine is a simple translation of the MINPACK lmder routine with minor changes. The changes include the over-determined resolution, the use of inherited convergence checker and the Q.R. decomposition which has been rewritten following the algorithm described in the P. Lascaux and R. Theodor book Analyse numérique matricielle appliquée à l'art de l'ingénieur, Masson 1986.

The authors of the original fortran version are:

  • Argonne National Laboratory. MINPACK project. March 1980
  • Burton S. Garbow
  • Kenneth E. Hillstrom
  • Jorge J. More
The redistribution policy for MINPACK is available here, for convenience, it is reproduced below.

Minpack Copyright Notice (1999) University of Chicago. All rights reserved
Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:
  1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
  2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
  3. The end-user documentation included with the redistribution, if any, must include the following acknowledgment: This product includes software developed by the University of Chicago, as Operator of Argonne National Laboratory. Alternately, this acknowledgment may appear in the software itself, if and wherever such third-party acknowledgments normally appear.
  4. WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS" WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4) DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL BE CORRECTED.
  5. LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT, INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE, EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE POSSIBILITY OF SUCH LOSS OR DAMAGES.
    Since:
    2.0
    • Field Details

      • solvedCols

        private int solvedCols
        Deprecated.
        Number of solved point.
      • diagR

        private double[] diagR
        Deprecated.
        Diagonal elements of the R matrix in the Q.R. decomposition.
      • jacNorm

        private double[] jacNorm
        Deprecated.
        Norms of the columns of the jacobian matrix.
      • beta

        private double[] beta
        Deprecated.
        Coefficients of the Householder transforms vectors.
      • permutation

        private int[] permutation
        Deprecated.
        Columns permutation array.
      • rank

        private int rank
        Deprecated.
        Rank of the jacobian matrix.
      • lmPar

        private double lmPar
        Deprecated.
        Levenberg-Marquardt parameter.
      • lmDir

        private double[] lmDir
        Deprecated.
        Parameters evolution direction associated with lmPar.
      • initialStepBoundFactor

        private final double initialStepBoundFactor
        Deprecated.
        Positive input variable used in determining the initial step bound.
      • costRelativeTolerance

        private final double costRelativeTolerance
        Deprecated.
        Desired relative error in the sum of squares.
      • parRelativeTolerance

        private final double parRelativeTolerance
        Deprecated.
        Desired relative error in the approximate solution parameters.
      • orthoTolerance

        private final double orthoTolerance
        Deprecated.
        Desired max cosine on the orthogonality between the function vector and the columns of the jacobian.
      • qrRankingThreshold

        private final double qrRankingThreshold
        Deprecated.
        Threshold for QR ranking.
      • weightedResidual

        private double[] weightedResidual
        Deprecated.
        Weighted residuals.
      • weightedJacobian

        private double[][] weightedJacobian
        Deprecated.
        Weighted Jacobian.
    • Constructor Details

      • LevenbergMarquardtOptimizer

        public LevenbergMarquardtOptimizer()
        Deprecated.
        Build an optimizer for least squares problems with default values for all the tuning parameters (see the other contructor. The default values for the algorithm settings are:
        • Initial step bound factor: 100
        • Cost relative tolerance: 1e-10
        • Parameters relative tolerance: 1e-10
        • Orthogonality tolerance: 1e-10
        • QR ranking threshold: Precision.SAFE_MIN
      • LevenbergMarquardtOptimizer

        public LevenbergMarquardtOptimizer(ConvergenceChecker<PointVectorValuePair> checker)
        Deprecated.
        Constructor that allows the specification of a custom convergence checker. Note that all the usual convergence checks will be disabled. The default values for the algorithm settings are:
        • Initial step bound factor: 100
        • Cost relative tolerance: 1e-10
        • Parameters relative tolerance: 1e-10
        • Orthogonality tolerance: 1e-10
        • QR ranking threshold: Precision.SAFE_MIN
        Parameters:
        checker - Convergence checker.
      • LevenbergMarquardtOptimizer

        public LevenbergMarquardtOptimizer(double initialStepBoundFactor, ConvergenceChecker<PointVectorValuePair> checker, double costRelativeTolerance, double parRelativeTolerance, double orthoTolerance, double threshold)
        Deprecated.
        Constructor that allows the specification of a custom convergence checker, in addition to the standard ones.
        Parameters:
        initialStepBoundFactor - Positive input variable used in determining the initial step bound. This bound is set to the product of initialStepBoundFactor and the euclidean norm of diag * x if non-zero, or else to initialStepBoundFactor itself. In most cases factor should lie in the interval (0.1, 100.0). 100 is a generally recommended value.
        checker - Convergence checker.
        costRelativeTolerance - Desired relative error in the sum of squares.
        parRelativeTolerance - Desired relative error in the approximate solution parameters.
        orthoTolerance - Desired max cosine on the orthogonality between the function vector and the columns of the Jacobian.
        threshold - Desired threshold for QR ranking. If the squared norm of a column vector is smaller or equal to this threshold during QR decomposition, it is considered to be a zero vector and hence the rank of the matrix is reduced.
      • LevenbergMarquardtOptimizer

        public LevenbergMarquardtOptimizer(double costRelativeTolerance, double parRelativeTolerance, double orthoTolerance)
        Deprecated.
        Build an optimizer for least squares problems with default values for some of the tuning parameters (see the other contructor. The default values for the algorithm settings are:
        Parameters:
        costRelativeTolerance - Desired relative error in the sum of squares.
        parRelativeTolerance - Desired relative error in the approximate solution parameters.
        orthoTolerance - Desired max cosine on the orthogonality between the function vector and the columns of the Jacobian.
      • LevenbergMarquardtOptimizer

        public LevenbergMarquardtOptimizer(double initialStepBoundFactor, double costRelativeTolerance, double parRelativeTolerance, double orthoTolerance, double threshold)
        Deprecated.
        The arguments control the behaviour of the default convergence checking procedure. Additional criteria can defined through the setting of a ConvergenceChecker.
        Parameters:
        initialStepBoundFactor - Positive input variable used in determining the initial step bound. This bound is set to the product of initialStepBoundFactor and the euclidean norm of diag * x if non-zero, or else to initialStepBoundFactor itself. In most cases factor should lie in the interval (0.1, 100.0). 100 is a generally recommended value.
        costRelativeTolerance - Desired relative error in the sum of squares.
        parRelativeTolerance - Desired relative error in the approximate solution parameters.
        orthoTolerance - Desired max cosine on the orthogonality between the function vector and the columns of the Jacobian.
        threshold - Desired threshold for QR ranking. If the squared norm of a column vector is smaller or equal to this threshold during QR decomposition, it is considered to be a zero vector and hence the rank of the matrix is reduced.
    • Method Details

      • doOptimize

        protected PointVectorValuePair doOptimize()
        Deprecated.
        Perform the bulk of the optimization algorithm.
        Specified by:
        doOptimize in class BaseAbstractMultivariateVectorOptimizer<DifferentiableMultivariateVectorFunction>
        Returns:
        the point/value pair giving the optimal value for the objective function.
      • determineLMParameter

        private void determineLMParameter(double[] qy, double delta, double[] diag, double[] work1, double[] work2, double[] work3)
        Deprecated.
        Determine the Levenberg-Marquardt parameter.

        This implementation is a translation in Java of the MINPACK lmpar routine.

        This method sets the lmPar and lmDir attributes.

        The authors of the original fortran function are:

        • Argonne National Laboratory. MINPACK project. March 1980
        • Burton S. Garbow
        • Kenneth E. Hillstrom
        • Jorge J. More

        Luc Maisonobe did the Java translation.

        Parameters:
        qy - array containing qTy
        delta - upper bound on the euclidean norm of diagR * lmDir
        diag - diagonal matrix
        work1 - work array
        work2 - work array
        work3 - work array
      • determineLMDirection

        private void determineLMDirection(double[] qy, double[] diag, double[] lmDiag, double[] work)
        Deprecated.
        Solve a*x = b and d*x = 0 in the least squares sense.

        This implementation is a translation in Java of the MINPACK qrsolv routine.

        This method sets the lmDir and lmDiag attributes.

        The authors of the original fortran function are:

        • Argonne National Laboratory. MINPACK project. March 1980
        • Burton S. Garbow
        • Kenneth E. Hillstrom
        • Jorge J. More

        Luc Maisonobe did the Java translation.

        Parameters:
        qy - array containing qTy
        diag - diagonal matrix
        lmDiag - diagonal elements associated with lmDir
        work - work array
      • qrDecomposition

        private void qrDecomposition(RealMatrix jacobian) throws ConvergenceException
        Deprecated.
        Decompose a matrix A as A.P = Q.R using Householder transforms.

        As suggested in the P. Lascaux and R. Theodor book Analyse numérique matricielle appliquée à l'art de l'ingénieur (Masson, 1986), instead of representing the Householder transforms with uk unit vectors such that:

         Hk = I - 2uk.ukt
         
        we use k non-unit vectors such that:
         Hk = I - betakvk.vkt
         
        where vk = ak - alphak ek. The betak coefficients are provided upon exit as recomputing them from the vk vectors would be costly.

        This decomposition handles rank deficient cases since the tranformations are performed in non-increasing columns norms order thanks to columns pivoting. The diagonal elements of the R matrix are therefore also in non-increasing absolute values order.

        Parameters:
        jacobian - Weighted Jacobian matrix at the current point.
        Throws:
        ConvergenceException - if the decomposition cannot be performed
      • qTy

        private void qTy(double[] y)
        Deprecated.
        Compute the product Qt.y for some Q.R. decomposition.
        Parameters:
        y - vector to multiply (will be overwritten with the result)