Levmarmethods

Extended_rate_function(opt, varargin)[source]
USAGE

[hxk, ghxk] = Extended_rate_function (opt, varargin)

INPUTS

  • xk – current point;

  • opt – structure includes required parameters:

    • .FR - concatenation of forward and reverse stoichiometric matrix

    • .A - Reduced forward stoichiometric matrix

    • .B - Reduced reverse stoichiometric matrix

    • .L - left null space of R-F

    • .l0 - positive initial concentration

    • .k - initial kinetic

OUTPUTS

  • hxk – the vector h(xk)

  • ghxk – gradient of h at xk

GLM_FY(mapp, lin_sym_solver, x0, options)[source]

GLM_FY is a Levenberg-Marquardt algorithm for solving systems of nonlinear equations \(h(x) = 0\), x in \(R^m\) using the nonlinear unconstrained minimization \(\textrm{min} \psi(x) = 1/2 ||h(x)||^2\) s.t. x in \(R^m\).

USAGE

[x_best, psi_best, out] = GLM_FY (mapp, lin_sym_solver, x0, options)

INPUTS

  • mapp – function handle provides h(x) and gradient h(x)

  • lin_sym_solver – function handle for solving the linear system

  • x0 – initial point

  • options – structure including the parameteres of scheme

    • .eta - parameter of the scheme

    • .MaxNumIter - maximum number of iterations

    • .MaxNumMapEval - maximum number of function evaluations

    • .MaxNumGmapEval - maximum number of subgradient evaluations

    • .TimeLimit - maximum running time

    • .epsilon - accuracy parameter

    • .x_opt - optimizer

    • .psi_opt - optimum

    • .adaptive - update lambda adaptively

    • .flag_x_error - 1: saves \(x_{error}\), 0: do not saves \(x_{error}\) (default)

    • .flag_psi_error - 1: saves \(\psi_{error}\), 0: do not saves \(\psi_{error}\) (default)

    • .flag_time - 1: saves \(\psi_{error}\), 0: do not saves \(\psi_{error}\) (default)

    • .Stopping_Crit - stopping criterion

      1. stop if \(||grad|| \leq \epsilon\)

      2. stop if \(||nhxk|| \leq \epsilon\)

      3. stop if MaxNumIter is reached

      4. stop if MaxNumMapEval is reached

      5. stop if MaxNumGmapEval is reached

      6. stop if TimeLimit is reached

      7. stop if \(||grad|| \leq \textrm{max}(\epsilon, \epsilon^2 * ngradx0)\)

      8. stop if \(||nhxk|| \leq \textrm{max}(\epsilon, \epsilon^2 * nhx0)\)

      9. stop if (default) \(||hxk|| \leq \epsilon\) or MaxNumIter is reached

OUTPUTS

  • x_best – the best approximation of the optimizer

  • psi_best – the best approximation of the optimum

  • out – structure including more output information

    • .T - running time

    • .Niter - total number of iterations

    • .Nmap - total number of mapping evaluations

    • .Ngmap - total number of mapping gradient evaluations

    • .merit_func - array including all merit function values

    • .x_error - relative error \(\textrm{norm}(x_k(:)-x_{opt}(:))/\textrm{norm}(x_{opt})\)

    • .psi_error - relative error \((\psi_k-\psi_{opt})/(\psi_0-\psi_{opt}))\)

    • .Status - reason of termination

GLM_YF(mapp, lin_sym_solver, x0, options)[source]

GLM_YF is a Levenberg-Marquardt algorithm for solving systems of nonlinear equations \(h(x) = 0\), x in \(R^m\) using the nonlinear unconstrained minimization \(\textrm{min} \psi(x) = 1/2 ||h(x)||^2\) s.t. x in \(R^m\).

USAGE

[x_best, psi_best, out]=GLM_YF (mapp, lin_sym_solver, x0, options)

INPUTS

  • mapp – function handle provides h(x) and gradient h(x)

  • lin_sym_solver – function handle for solving the linear system

  • x0 – initial point

  • options – structure including the parameteres of scheme

    • .eta - parameter of the scheme

    • .MaxNumIter - maximum number of iterations

    • .MaxNumMapEval - maximum number of function evaluations

    • .MaxNumGmapEval - maximum number of subgradient evaluations

    • .TimeLimit - maximum running time

    • .epsilon - accuracy parameter

    • .x_opt - optimizer

    • .psi_opt - optimum

    • .adaptive - update lambda adaptively

    • .flag_x_error - 1: saves \(x_{error}\), 0: do not saves \(x_{error}\) (default)

    • .flag_psi_error - 1: saves \(\psi_{error}\), 0: do not saves \(\psi_{error}\) (default)

    • .flag_time - 1: saves \(\psi_{error}\), 0: do not saves \(\psi_{error}\) (default)

    • .Stopping_Crit - stopping criterion

      1. stop if \(||grad|| \leq \epsilon\)

      2. stop if \(||nhxk|| \leq \epsilon\)

      3. stop if MaxNumIter is reached

      4. stop if MaxNumMapEval is reached

      5. stop if MaxNumGmapEval is reached

      6. stop if TimeLimit is reached

      7. stop if \(||grad|| \leq \textrm{max}(\epsilon, \epsilon^2 * ngradx0)\)

      8. stop if \(||nhxk|| \leq \textrm{max}(\epsilon, \epsilon^2 * nhx0)\)

      9. stop if (default) \(||hxk|| \leq \epsilon\) or MaxNumIter is reached

OUTPUT

  • x_best – the best approximation of the optimizer

  • psi_best – the best approximation of the optimum

  • out – structure including more output information

    • .T - running time

    • .Niter - total number of iterations

    • .Nmap - total number of mapping evaluations

    • .Ngmap - total number of mapping gradient evaluations

    • .merit_func - array including all merit function values

    • .x_error - relative error \(\textrm{norm}(x_k(:)-x_{opt}(:))/\textrm{norm}(x_{opt})\)

    • .psi_error - relative error \((\psi_k-\psi_{opt})/(\psi_0-\psi_{opt}))\)

    • .Status - reason of termination

Initialization(options)[source]

Initialization is a function for initializing the parameters of LLM and GLM. If some parameters specified by the user Initialization uses these parameters. Otherwise, the default values will be employed.

USAGE

[eta, epsilon, MaxNumIter, MaxNumMapEval, MaxNumGmapEval, adaptive, TimeLimit, flag_x_error, flag_psi_error, flag_time, Stopping_Crit] = Initialization (options)

INPUTS

options – structure including the parameteres of schemes

  • .eta - parameter of the scheme

  • .MaxNumIter - maximum number of iterations

  • .MaxNumMapEval - maximum number of function evaluations

  • .MaxNumGmapEval - maximum number of subgradient evaluations

  • .TimeLimit - maximum running time

  • .epsilon - accuracy parameter

  • .x_opt - optimizer

  • .psi_opt - optimum

  • .adaptive - update lambda adaptively

  • .flag_x_error - 1: saves \(x_{error}\), 0: do not saves \(x_{error}\) (default)

  • .flag_psi_error - 1: saves \(\psi_{error}\), 0: do not saves \(\psi_{error}\) (default)

  • .flag_time - 1: saves \(\psi_{error}\), 0: do not saves \(\psi_{error}\) (default)

  • .Stopping_Crit - stopping criterion

    1. stop if \(||grad|| \leq \epsilon\)

    2. stop if \(||nhxk|| \leq \epsilon\)

    3. stop if MaxNumIter is reached

    4. stop if MaxNumMapEval is reached

    5. stop if MaxNumGmapEval is reached

    6. stop if TimeLimit is reached

    7. stop if \(||grad|| \leq \textrm{max}(\epsilon, \epsilon^2 * ngradx0)\)

    8. stop if \(||nhxk|| \leq \textrm{max}(\epsilon, \epsilon^2 * nhx0)\)

    9. stop if (default) \(||hxk|| \leq \epsilon\) or MaxNumIter is reached

OUTPUTS

  • eta – parameter of the scheme

  • MaxNumIter – maximum number of iterations

  • MaxNumMapEval – maximum number of function evaluations

  • MaxNumGmapEval – maximum number of subgradient evaluations

  • TimeLimit – maximum running time

  • epsilon – accuracy parameter

  • x_opt – optimizer

  • psi_opt – optimum

  • adaptive – update lambda adaptively

  • flag_x_error – 1: saves \(x_{error}\), 0: do not saves \(x_{error}\) (default)

  • flag_psi_error – 1: saves \(\psi_{error}\), 0: do not saves \(\psi_{error}\) (default)

  • flag_time – 1: saves \(\psi_{error}\), 0: do not saves \(\psi_{error}\) (default)

  • Stopping_Crit – stopping criterion

LLM(mapp, lin_sym_solver, x0, options)[source]

LLM is a Levenberg-Marquardt algorithm for solving systems of nonlinear equations \(h(x) = 0\), x in \(R^m\) using the nonlinear unconstrained minimization \(\textrm{min} \psi(x) = 1/2 ||h(x)||^2\) s.t. x in \(R^m\).

USAGE

[x_best, psi_best, out] = LLM (mapp, lin_sym_solver, x0, options)

INPUTS

  • mapp – function handle provides h(x) and gradient h(x)

  • lin_sym_solver – function handle for solving the linear system

  • x0 – initial point

  • options – structure including the parameteres of scheme

    • .eta - parameter of the scheme

    • .MaxNumIter - maximum number of iterations

    • .MaxNumMapEval - maximum number of function evaluations

    • .MaxNumGmapEval - maximum number of subgradient evaluations

    • .TimeLimit - maximum running time

    • .epsilon - accuracy parameter

    • .x_opt - optimizer

    • .psi_opt - optimum

    • .adaptive - update lambda adaptively

    • .flag_x_error - 1: saves \(x_{error}\), 0: do not saves \(x_{error}\) (default)

    • .flag_psi_error - 1: saves \(\psi_{error}\), 0: do not saves \(\psi_{error}\) (default)

    • .flag_time - 1: saves \(\psi_{error}\), 0: do not saves \(\psi_{error}\) (default)

    • .Stopping_Crit - stopping criterion

      1. stop if \(||grad|| \leq \epsilon\)

      2. stop if \(||nhxk|| \leq \epsilon\)

      3. stop if MaxNumIter is reached

      4. stop if MaxNumMapEval is reached

      5. stop if MaxNumGmapEval is reached

      6. stop if TimeLimit is reached

      7. stop if \(||grad|| \leq \textrm{max}(\epsilon, \epsilon^2 * ngradx0)\)

      8. stop if \(||nhxk|| \leq \textrm{max}(\epsilon, \epsilon^2 * nhx0)\)

      9. stop if (default) \(||hxk|| \leq \epsilon\) or MaxNumIter is reached

OUTPUTS

  • x_best – the best approximation of the optimizer

  • psi_best – the best approximation of the optimum

  • out – structure including more output information

    • .T - running time

    • .Niter - total number of iterations

    • .Nmap - total number of mapping evaluations

    • .Ngmap - total number of mapping gradient evaluations

    • .merit_func - array including all merit function values

    • .x_error - relative error \(\textrm{norm}(x_k(:)-x_{opt}(:))/\textrm{norm}(x_{opt})\)

    • .psi_error - relative error \((\psi_k-\psi_{opt})/(\psi_0-\psi_{opt}))\)

    • .Status - reason of termination

LMLS(mapp, lin_sym_solver, x0, options)[source]

LMLS is a Levenberg-Marquardt algorithm for solving systems of nonlinear equations \(h(x) = 0\), x in \(R^m\) using the nonlinear unconstrained minimization \(\textrm{min} \psi(x) = 1/2 ||h(x)||^2\) s.t. x in \(R^m\).

USAGE

[x_best, psi_best, out] = LMLS (mapp, lin_sym_solver, x0, options)

INPUTS

  • mapp – function handle provides h(x) and gradient h(x)

  • lin_sym_solver – function handle for solving the linear system

  • x0 – initial point

  • options – structure including the parameteres of scheme

    • .eta - parameter of the scheme

    • .MaxNumIter - maximum number of iterations

    • .MaxNumMapEval - maximum number of function evaluations

    • .MaxNumGmapEval - maximum number of subgradient evaluations

    • .TimeLimit - maximum running time

    • .epsilon - accuracy parameter

    • .x_opt - optimizer

    • .psi_opt - optimum

    • .adaptive - update lambda adaptively

    • .flag_x_error - 1: saves \(x_{error}\), 0: do not saves \(x_{error}\) (default)

    • .flag_psi_error - 1: saves \(\psi_{error}\), 0: do not saves \(\psi_{error}\) (default)

    • .flag_time - 1: saves \(\psi_{error}\), 0: do not saves \(\psi_{error}\) (default)

    • .Stopping_Crit - stopping criterion

      1. stop if \(||grad|| \leq \epsilon\)

      2. stop if \(||nhxk|| \leq \epsilon\)

      3. stop if MaxNumIter is reached

      4. stop if MaxNumMapEval is reached

      5. stop if MaxNumGmapEval is reached

      6. stop if TimeLimit is reached

      7. stop if \(||grad|| \leq \textrm{max}(\epsilon, \epsilon^2 * ngradx0)\)

      8. stop if \(||nhxk|| \leq \textrm{max}(\epsilon, \epsilon^2 * nhx0)\)

      9. stop if (default) \(||hxk|| \leq \epsilon\) or MaxNumIter is reached

OUTPUTS

  • x_best – the best approximation of the optimizer

  • psi_best – the best approximation of the optimum

  • out – structure including more output information

    • .T - running time

    • .Niter - total number of iterations

    • .Nmap - total number of mapping evaluations

    • .Ngmap - total number of mapping gradient evaluations

    • .merit_func - array including all merit function values

    • .x_error - relative error \(\textrm{norm}(x_k(:)-x_{opt}(:))/\textrm{norm}(x_{opt})\)

    • .psi_error - relative error \((\psi_k-\psi_{opt})/(\psi_0-\psi_{opt}))\)

    • .Status - reason of termination

LMTR(mapp, lin_sym_solver, x0, options)[source]

LMTR is a Levenberg-Marquardt algorithm for solving systems of nonlinear equations \(h(x) = 0\), x in \(R^m\) using the nonlinear unconstrained minimization \(\textrm{min} \psi(x) = 1/2 ||h(x)||^2\) s.t. x in \(R^m\).

USAGE

[x_best, psi_best, out] = LMTR (mapp, lin_sym_solver, x0, options)

INPUTS

  • mapp – function handle provides h(x) and gradient h(x)

  • lin_sym_solver – function handle for solving the linear system

  • x0 – initial point

  • options – structure including the parameteres of scheme

    • .eta - parameter of the scheme

    • .MaxNumIter - maximum number of iterations

    • .MaxNumMapEval - maximum number of function evaluations

    • .MaxNumGmapEval - maximum number of subgradient evaluations

    • .TimeLimit - maximum running time

    • .epsilon - accuracy parameter

    • .x_opt - optimizer

    • .psi_opt - optimum

    • .adaptive - update lambda adaptively

    • .flag_x_error - 1: saves \(x_{error}\), 0: do not saves \(x_{error}\) (default)

    • .flag_psi_error - 1: saves \(\psi_{error}\), 0: do not saves \(\psi_{error}\) (default)

    • .flag_time - 1: saves \(\psi_{error}\), 0: do not saves \(\psi_{error}\) (default)

    • .Stopping_Crit - stopping criterion

      1. stop if \(||grad|| \leq \epsilon\)

      2. stop if \(||nhxk|| \leq \epsilon\)

      3. stop if MaxNumIter is reached

      4. stop if MaxNumMapEval is reached

      5. stop if MaxNumGmapEval is reached

      6. stop if TimeLimit is reached

      7. stop if \(||grad|| \leq \textrm{max}(\epsilon, \epsilon^2 * ngradx0)\)

      8. stop if \(||nhxk|| \leq \textrm{max}(\epsilon, \epsilon^2 * nhx0)\)

      9. stop if (default) \(||hxk|| \leq \epsilon\) or MaxNumIter is reached

OUTPUTS

  • x_best – the best approximation of the optimizer

  • psi_best – the best approximation of the optimum

  • out – structure including more output information

    • .T - running time

    • .Niter - total number of iterations

    • .Nmap - total number of mapping evaluations

    • .Ngmap - total number of mapping gradient evaluations

    • .merit_func - array including all merit function values

    • .x_error - relative error \(\textrm{norm}(x_k(:)-x_{opt}(:))/\textrm{norm}(x_{opt})\)

    • .psi_error - relative error \((\psi_k-\psi_{opt})/(\psi_0-\psi_{opt}))\)

    • .Status - reason of termination

LevMar(mapp, lin_sym_solver, x0, options)[source]

LevMar is a Levenberg-Marquardt trust-region algorithm for solving systems of nonlinear equations \(h(x) = 0\), x in \(R^m\) using the nonlinear unconstrained minimization \(\textrm{min}\ \psi(x) = 1/2 ||h(x)||^2\) s.t. x in \(R^m\).

USAGE

[x_best, psi_best, out] = LevMar (mapp, lin_sym_solver, x0, options)

INPUTS

  • mapp – function handle provides h(x) and gradient h(x)

  • lin_sym_solver – function handle for solving the linear system

  • x0 – initial point

  • options – structure including the parameteres of scheme

    • .eta - parameter of the scheme

    • .MaxNumIter - maximum number of iterations

    • .MaxNumMapEval - maximum number of function evaluations

    • .MaxNumGmapEval - maximum number of subgradient evaluations

    • .TimeLimit - maximum running time

    • .epsilon - accuracy parameter

    • .x_opt - optimizer

    • .psi_opt - optimum

    • .adaptive - update lambda adaptively

    • .flag_x_error - 1: saves \(x_{error}\), 0: do not saves \(x_{error}\) (default)

    • .flag_psi_error - 1: saves \(\psi_{error}\), 0: do not saves \(\psi_{error}\) (default)

    • .flag_time - 1: saves \(\psi_{error}\), 0: do not saves \(\psi_{error}\) (default)

    • .Stopping_Crit - stopping criterion

      1. stop if \(||grad|| \leq \epsilon\)

      2. stop if \(||nhxk|| \leq \epsilon\)

      3. stop if MaxNumIter is reached

      4. stop if MaxNumMapEval is reached

      5. stop if MaxNumGmapEval is reached

      6. stop if TimeLimit is reached

      7. stop if \(||grad|| \leq \textrm{max}(\epsilon, \epsilon^2 * ngradx0)\)

      8. stop if \(||nhxk|| \leq \textrm{max}(\epsilon, \epsilon^2 * nhx0)\)

      9. stop if (default) \(||hxk|| \leq \epsilon\) or MaxNumIter is reached

OUTPUT

  • x_best – the best approximation of the optimizer

  • psi_best – the best approximation of the optimum

  • out – structure including more output information

    • .T - running time

    • .Niter - total number of iterations

    • .Nmap - total number of mapping evaluations

    • .Ngmap - total number of mapping gradient evaluations

    • .merit_func - array including all merit function values

    • .x_error - relative error \(\textrm{norm}(x_k(:)-x_{opt}(:))/\textrm{norm}(x_{opt})\)

    • .psi_error - relative error \((\psi_k-\psi_{opt})/(\psi_0-\psi_{opt}))\)

    • .Status - reason of termination

Rate_function1(opt, varargin)[source]
USAGE

[fxk, gfxk] = Rate_function1 (opt, varargin)

INPUTS

  • xk – current point;

  • opt – structure includes required parameters;

    • .FR - concatenation of forward and reverse stoichiometric matrix

    • .A - Reduced forward stoichiometric matrix

    • .B - Reduced reverse stoichiometric matrix

    • .k - initial kinetic

OUTPUTS

  • hxk – the vector h(xk)

  • ghxk – gradient of h at xk

StopCriterion(grad, nhxk, Niter, Nmap, Ngmap, MaxNumIter, MaxNumMapEval, MaxNumGmapEval, T, TimeLimit, epsilon, nhx0, ngradx0, Stopping_Crit)[source]

StopCriterion is a function checking that one of the stopping criteria holds to terminate LLM and GLM. It perepare the status determining why the algorithm is stopped.

USAGE

[StopFlag, Status] = StopCriterion (grad, nhxk, Niter, Nmap, Ngmap, MaxNumIter, MaxNumMapEval, MaxNumGmapEval, T, TimeLimit, epsilon, nhx0, ngradx0, Stopping_Crit)

INPUTS

  • grad – gradient of the merit funcrion

  • nhxk – the norm 2 of h(xk)

  • MaxNumIter – maximum number of iterations

  • MaxNumMapEval – maximum number of function evaluations

  • MaxNumGmapEval – maximum number of subgradient evaluations

  • TimeLimit – maximum running time

  • epsilon – accuracy parameter

  • Stopping_Crit – stopping criterion

    1. stop if \(||grad|| \leq \epsilon\)

    2. stop if \(||nhxk|| \leq \epsilon\)

    3. stop if MaxNumIter is reached

    4. stop if MaxNumMapEval is reached

    5. stop if MaxNumGmapEval is reached

    6. stop if TimeLimit is reached

    7. stop if \(||grad|| \leq \textrm{max}(\epsilon, \epsilon^2 * ngradx0)\)

    8. stop if \(||nhxk|| \leq \textrm{max}(\epsilon, \epsilon^2 * nhx0)\)

    9. stop if (default) \(||hxk|| \leq \epsilon\) or MaxNumIter is reached

OUTPUTS

  • StopFlag – 1: if one of the stopping criteria holds, 0: if none of the stopping criteria holds

  • Status – the reason of the scheme termination

lin_sym_solver_mldivide(Hk, grad)[source]

Solves the linear system \(Hkdk = -grad\).

USAGE

dk = lin_sym_solver_mldivide (Hk, grad)

INPUTS

  • ghxk – gradient of h at xk

  • grad – gradient of the merit function at xk

  • muk – the parameter muk

OUTPUT

dk – the solution of the linear system