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NAME

       libdogleg - A general purpose optimizer to solve data fitting problems

NOTICE

       If you're considering using this library for new projects, please look at the "ceres"
       solver instead:

       <http://ceres-solver.org/>

       "libdogleg" is not deprecated, and I will fix bugs as they're reported. However "ceres" is
       more feature-rich and much more widely used. I consider "ceres" to be a superset of
       "libdogleg", and if "ceres" was available when I wrote "libdogleg", I would not have
       written it.

DESCRIPTION

       This is a library for solving large-scale nonlinear optimization problems. By employing
       sparse linear algebra, it is taylored for problems that have weak coupling between the
       optimization variables. For appropriately sparse problems this results in massive
       performance gains.

       For smaller problems with dense Jacobians a dense mode is available also. This utilizes
       the same optimization loop as the sparse code, but uses dense linear algebra.

       The main task of this library is to find the vector p that minimizes

       norm2( x )

       where x = f(p) is a vector that has higher dimensionality than p. The user passes in a
       callback function (of type "dogleg_callback_t") that takes in the vector p and returns the
       vector x and a matrix of derivatives J = df/dp. J is a matrix with a row for each element
       of f and a column for each element of p. If J is a sparse matrix, then this library can
       take advantage of that, which results in substantial increases in computational efficiency
       if most entries of J are 0. J is stored row-first in the callback routine. libdogleg uses
       a column-first data representation so it references the transpose of J (called Jt). J
       stored row-first is identical to Jt stored column-first; this is purely a naming choice.

       This library implements Powell's dog-leg algorithm to solve the problem. Like the more-
       widely-known Levenberg-Marquardt algorithm, Powell's dog-leg algorithm solves a nonlinear
       optimization problem by interpolating between a Gauss-Newton step and a gradient descent
       step. Improvements over LM are

       •   a more natural representation of the linearity of the operating point (trust region
           size vs a vague lambda term).

       •   significant efficiency gains, since a matrix inversion isn't needed to retry a
           rejected step

       The algorithm is described in many places, originally in

       M. Powell. A Hybrid Method for Nonlinear Equations. In P. Rabinowitz, editor, Numerical
       Methods for Nonlinear Algebraic Equations, pages 87-144.  Gordon and Breach Science,
       London, 1970.

       Various enhancements to Powell's original method are described in the literature; at this
       time only the original algorithm is implemented here.

       The sparse matrix algebra is handled by the CHOLMOD library, written by Tim Davis. Parts
       of CHOLMOD are licensed under the GPL and parts under the LGPL. Only the LGPL pieces are
       used here, allowing libdogleg to be licensed under the LGPL as well. Due to this I lose
       some convenience (all simple sparse matrix arithmetic in CHOLMOD is GPL-ed) and some
       performance (the fancier computational methods, such as supernodal analysis are GPL-ed).
       For my current applications the performance losses are minor.

FUNCTIONS AND TYPES

   Main API
       dogleg_optimize

       This is the main call to the library for sparse Jacobians. Its declared as

        double dogleg_optimize(double* p, unsigned int Nstate,
                               unsigned int Nmeas, unsigned int NJnnz,
                               dogleg_callback_t* f, void* cookie,
                               dogleg_solverContext_t** returnContext);

       •   p is the initial estimate of the state vector (and holds the final result)

       •   "Nstate" specifies the number of optimization variables (elements of p)

       •   "Nmeas" specifies the number of measurements (elements of x). "Nmeas >= Nstate" is a
           requirement

       •   "NJnnz" specifies the number of non-zero elements of the jacobian matrix df/dp. In a
           dense matrix "Jnnz = Nstate*Nmeas". We are dealing with sparse jacobians, so "NJnnz"
           should be far less. If not, libdogleg is not an appropriate routine to solve this
           problem.

       •   "f" specifies the callback function that the optimization routine calls to sample the
           problem being solved

       •   "cookie" is an arbitrary data pointer passed untouched to "f"

       •   If not "NULL", "returnContext" can be used to retrieve the full context structure from
           the solver. This can be useful since this structure contains the latest operating
           point values. It also has an active "cholmod_common" structure, which can be reused if
           more CHOLMOD routines need to be called externally. You usually want
           "returnContext->beforeStep". If this data is requested, the user is required to free
           it with "dogleg_freeContext" when done.

       "dogleg_optimize" returns norm2( x ) at the minimum, or a negative value if an error
       occurred.

       dogleg_optimize_dense

       This is the main call to the library for dense Jacobians. Its declared as

        double dogleg_optimize_dense(double* p, unsigned int Nstate,
                                     unsigned int Nmeas,
                                     dogleg_callback_dense_t* f, void* cookie,
                                     dogleg_solverContext_t** returnContext);

       The arguments are almost identical to those in the "dogleg_optimize" call.

       •   p is the initial estimate of the state vector (and holds the final result)

       •   "Nstate" specifies the number of optimization variables (elements of p)

       •   "Nmeas" specifies the number of measurements (elements of x). "Nmeas >= Nstate" is a
           requirement

       •   "f" specifies the callback function that the optimization routine calls to sample the
           problem being solved. Note that this callback has a different type from that in
           "dogleg_optimize"

       •   "cookie" is an arbitrary data pointer passed untouched to "f"

       •   If not "NULL", "returnContext" can be used to retrieve the full context structure from
           the solver. This can be useful since this structure contains the latest operating
           point values. You usually want "returnContext->beforeStep". If this data is requested,
           the user is required to free it with "dogleg_freeContext" when done.

       "dogleg_optimize" returns norm2( x ) at the minimum, or a negative value if an error
       occurred.

       dogleg_freeContext

       Used to deallocate memory used for an optimization cycle. Defined as:

        void dogleg_freeContext(dogleg_solverContext_t** ctx);

       If a pointer to a context is not requested (by passing "returnContext = NULL" to
       "dogleg_optimize"), libdogleg calls this routine automatically. If the user did retrieve
       this pointer, though, it must be freed with "dogleg_freeContext" when the user is
       finished.

       dogleg_computeJtJfactorization

       Computes the cholesky decomposition of JtJ. This function is only exposed if you need to
       touch libdogleg internals via returnContext. Sometimes after computing an optimization you
       want to do stuff with the factorization of JtJ, and this call ensures that the
       factorization is available. Most people don't need this function. If the comment wasn't
       clear, you don't need this function.

       This is declared as

        void dogleg_computeJtJfactorization(dogleg_operatingPoint_t* point,
                                            dogleg_solverContext_t* ctx);

       The arguments are

       •   "point" is the operating point of the system. Generally this will be
           "returnContext->beforeStep" where "returnContext" is from one of the
           "dogleg_optimize_..." functions.

       •   "ctx" is the dogleg context. Generally this will be "returnContext" from one of the
           "dogleg_optimize_..." functions

       dogleg_testGradient

       libdogleg requires the user to compute the jacobian matrix J. This is a performance
       optimization, since J could be computed by differences of x. This optimization is often
       worth the extra effort, but it creates a possibility that J will have a mistake and J =
       df/dp would not be true. To find these types of issues, the user can call

        void dogleg_testGradient(unsigned int var, const double* p0,
                                 unsigned int Nstate, unsigned int Nmeas, unsigned int NJnnz,
                                 dogleg_callback_t* f, void* cookie);

       This function computes the jacobian with center differences and compares the result with
       the jacobian computed by the callback function. It is recommended to do this for every
       variable while developing the program that uses libdogleg.

       •   "var" is the index of the variable being tested

       •   "p0" is the state vector p where we're evaluating the jacobian

       •   "Nstate", "Nmeas", "NJnnz" are the number of state variables, measurements and non-
           zero jacobian elements, as before

       •   "f" is the callback function, as before

       •   "cookie" is the user data, as before

       This function returns nothing, but prints out the test results.

       dogleg_testGradient_dense

       Very similar to "dogleg_testGradient", but for dense jacobians.

        void dogleg_testGradient_dense(unsigned int var, const double* p0,
                                       unsigned int Nstate, unsigned int Nmeas,
                                       dogleg_callback_dense_t* f, void* cookie);

       This function computes the jacobian with center differences and compares the result with
       the jacobian computed by the callback function. It is recommended to do this for every
       variable while developing the program that uses libdogleg.

       •   "var" is the index of the variable being tested

       •   "p0" is the state vector p where we're evaluating the jacobian

       •   "Nstate", "NJnnz" are the number of state variables, measurements

       •   "f" is the callback function, as before

       •   "cookie" is the user data, as before

       This function returns nothing, but prints out the test results.

       dogleg_callback_t

       The main user callback that specifies the sparse optimization problem has type

        typedef void (dogleg_callback_t)(const double*   p,
                                         double*         x,
                                         cholmod_sparse* Jt,
                                         void*           cookie);

       •   p is the current state vector

       •   x is the resulting f(p)

       •   Jt is the transpose of df/dp at p. As mentioned previously, Jt is stored column-first
           by CHOLMOD, which can be interpreted as storing J row-first by the user-defined
           callback routine

       •   The "cookie" is the user-defined arbitrary data passed into "dogleg_optimize".

       dogleg_callback_dense_t

       The main user callback that specifies the dense optimization problem has type

        typedef void (dogleg_callback_dense_t)(const double*   p,
                                               double*         x,
                                               double*         J,
                                               void*           cookie);

       •   p is the current state vector

       •   x is the resulting f(p)

       •   J is df/dp at p. J is stored row-first, with all the derivatives for the first
           measurement, then all the derivatives for the second measurement and so on.

       •   The "cookie" is the user-defined arbitrary data passed into "dogleg_optimize".

       dogleg_solverContext_t

       This is the solver context that can be retrieved through the "returnContext" parameter of
       the "dogleg_optimize" call. This structure contains all the internal state used by the
       solver. If requested, the user is responsible for calling "dogleg_freeContext" when done.
       This structure is defined as:

        typedef struct
        {
          cholmod_common  common;

          union
          {
            dogleg_callback_t*       f;
            dogleg_callback_dense_t* f_dense;
          };
          void*              cookie;

          // between steps, beforeStep contains the operating point of the last step.
          // afterStep is ONLY used while making the step. Externally, use beforeStep
          // unless you really know what you're doing
          dogleg_operatingPoint_t* beforeStep;
          dogleg_operatingPoint_t* afterStep;

          // The result of the last JtJ factorization performed. Note that JtJ is not
          // necessarily factorized at every step, so this is NOT guaranteed to contain
          // the factorization of the most recent JtJ
          union
          {
            cholmod_factor* factorization;

            // This is a factorization of JtJ, stored as a packed symmetric matrix
            // returned by dpptrf('L', ...)
            double*         factorization_dense;
          };

          // Have I ever seen a singular JtJ? If so, I add this constant to the diagonal
          // from that point on. This is a simple and fast way to deal with
          // singularities. This constant starts at 0, and is increased every time a
          // singular JtJ is encountered. This is suboptimal but works for me for now.
          double                   lambda;

          // Are we using sparse math (cholmod)?
          int                      is_sparse;
          int                      Nstate, Nmeasurements;
        } dogleg_solverContext_t;

       Some of the members are copies of the data passed into "dogleg_optimize"; some others are
       internal state. Of potential interest are

       •   "common" is a cholmod_common structure used by all CHOLMOD calls. This can be used for
           any extra CHOLMOD work the user may want to do

       •   "beforeStep" contains the operating point of the optimum solution. The user can
           analyze this data without the need to re-call the callback routine.

       dogleg_operatingPoint_t

       An operating point of the solver. This is a part of "dogleg_solverContext_t".  Various
       variables describing the operating point such as p, J, x, norm2(x) and Jt x are available.
       All of the just-mentioned variables are computed at every step and are thus always valid.

        // an operating point of the solver
        typedef struct
        {
          double*         p;
          double*         x;
          double          norm2_x;
          union
          {
            cholmod_sparse* Jt;
            double*         J_dense; // row-first: grad0, grad1, grad2, ...
          };
          double*         Jt_x;

          // the cached update vectors. It's useful to cache these so that when a step is rejected, we can
          // reuse these when we retry
          double*        updateCauchy;
          union
          {
            cholmod_dense* updateGN_cholmoddense;
            double*        updateGN_dense;
          };
          double         updateCauchy_lensq, updateGN_lensq; // update vector lengths

          // whether the current update vectors are correct or not
          int updateCauchy_valid, updateGN_valid;

          int didStepToEdgeOfTrustRegion;
        } dogleg_operatingPoint_t;

   Parameters
       It is not required to call any of these, but it's highly recommended to set the initial
       trust-region size and the termination thresholds to match the problem being solved.
       Furthermore, it's highly recommended for the problem being solved to be scaled so that
       every state variable affects the objective norm2( x ) equally. This makes this method's
       concept of "trust region" much more well-defined and makes the termination criteria work
       correctly.

       dogleg_setMaxIterations

       To set the maximum number of solver iterations, call

        void dogleg_setMaxIterations(int n);

       dogleg_setDebug

       To turn on debug output, call

        void dogleg_setDebug(int debug);

       with a non-zero value for "debug". By default, debug output is disabled.

       dogleg_setInitialTrustregion

       The optimization method keeps track of a trust region size. Here, the trust region is a
       ball in R^Nstate. When the method takes a step p -> p + delta_p, it makes sure that

       sqrt( norm2( delta_p ) ) < trust region size.

       The initial value of the trust region size can be set with

        void dogleg_setInitialTrustregion(double t);

       The dogleg algorithm is efficient when recomputing a rejected step for a smaller trust
       region, so set the initial trust region size to a value larger to a reasonable estimate;
       the method will quickly shrink the trust region to the correct size.

       dogleg_setThresholds

       The routine exits when the maximum number of iterations is exceeded, or a termination
       threshold is hit, whichever happens first. The termination thresholds are all designed to
       trigger when very slow progress is being made. If all went well, this slow progress is due
       to us finding the optimum. There are 3 termination thresholds:

       •   The function being minimized is E = norm2( x ) where x = f(p).

           dE/dp = 2*Jt*x where Jt is transpose(dx/dp).

            if( for every i  fabs(Jt_x[i]) < JT_X_THRESHOLD )
            { we are done; }

       •   The method takes discrete steps: p -> p + delta_p

            if( for every i  fabs(delta_p[i]) < UPDATE_THRESHOLD)
            { we are done; }

       •   The method dynamically controls the trust region.

            if(trustregion < TRUSTREGION_THRESHOLD)
            { we are done; }

       To set these threholds, call

        void dogleg_setThresholds(double Jt_x, double update, double trustregion);

       To leave a particular threshold alone, specify a negative value.

       dogleg_setTrustregionUpdateParameters

       This function sets the parameters that control when and how the trust region is updated.
       The default values should work well in most cases, and shouldn't need to be tweaked.

       Declaration looks like

        void dogleg_setTrustregionUpdateParameters(double downFactor, double downThreshold,
                                                   double upFactor,   double upThreshold);

       To see what the parameters do, look at "evaluateStep_adjustTrustRegion" in the source.
       Again, these should just work as is.

BUGS

       The current implementation doesn't handle a singular JtJ gracefully (JtJ = Jt * J).
       Analytically, JtJ is at worst positive semi-definite (has 0 eigenvalues). If a singular
       JtJ is ever encountered, from that point on, JtJ + lambda*I is inverted instead for some
       positive constant lambda. This makes the matrix strictly positive definite, but is sloppy.
       At least I should vary lambda. In my current applications, a singular JtJ only occurs if
       at a particular operating point the vector x has no dependence at all on some elements of
       p. In the general case other causes could exist, though.

       There's an inefficiency in that the callback always returns x and J. When I evaluate and
       reject a step, I do not end up using J at all. Dependng on the callback function, it may
       be better to ask for x and then, if the step is accepted, to ask for J.

AUTHOR

       Dima Kogan, "<dima@secretsauce.net>"

LICENSE AND COPYRIGHT

       Copyright 2011 Oblong Industries
                 2017 Dima Kogan <dima@secretsauce.net>

       This program is free software: you can redistribute it and/or modify it under the terms of
       the GNU Lesser General Public License as published by the Free Software Foundation, either
       version 3 of the License, or (at your option) any later version.

       This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY;
       without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
       See the GNU Lesser General Public License for more details.

       The full text of the license is available at <http://www.gnu.org/licenses>