Derfreemethods

BDF(mapp, x0, options)[source]

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

USAGE

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

INPUTS

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

  • x0 – initial point

  • options – structure including the parameteres of scheme

    • .MaxNumIter - maximum number of iterations

    • .MaxNumMapEval - maximum number of function evaluations

    • .TimeLimit - maximum running time

    • .epsilon - accuracy parameter

    • .x_opt - optimizer

    • .psi_opt - optimum

    • .alpha - constant with \(\alpha < 2 \sigma\)

    • .beta - reduction constant of the line search

    • .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 \(||nfxk|| \leq \epsilon\)

      2. stop if MaxNumIter is reached

      3. stop if MaxNumMapEval is reached

      4. stop if TimeLimit is reached

      5. 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

    • .merit_func - array including all merit function values

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

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

    • .Status - reason of termination

CSDF(mapp, x0, options)[source]

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

USAGE

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

INPUTS

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

  • x0 – initial point

  • options – structure including the parameteres of scheme

    • .MaxNumIter - maximum number of iterations

    • .MaxNumMapEval - maximum number of function evaluations

    • .TimeLimit - maximum running time

    • .epsilon - accuracy parameter

    • .x_opt - optimizer

    • .psi_opt - optimum

    • .sigma - strong duplomonotone parameter

    • .l - Lipschitz continuity constant of f

    • .tauBar - a constant for determining the step-size

    • .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 \(||nfxk|| \leq \epsilon\)

      2. stop if MaxNumIter is reached

      3. stop if MaxNumMapEval is reached

      4. stop if TimeLimit is reached

      5. 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

    • .merit_func - array including all merit function values

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

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

    • .Status - reason of termination

DBDF(mapp, x0, options)[source]

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

USAGE

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

INPUTS

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

  • x0 – initial point

  • options – structure including the parameteres of scheme

    • .MaxNumIter - maximum number of iterations

    • .MaxNumMapEval - maximum number of function evaluations

    • .TimeLimit - maximum running time

    • .epsilon - accuracy parameter

    • .x_opt - optimizer

    • .psi_opt - optimum

    • .alpha - constant with \(\alpha < 2 \sigma\)

    • .beta - reduction constant of the line search

    • .lambda_min - lower bound of the step-size

    • .lambda_max - upper bound of the step-size

    • .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 \(||nfxk|| \leq \epsilon\)

      2. stop if MaxNumIter is reached

      3. stop if MaxNumMapEval is reached

      4. stop if TimeLimit is reached

      5. 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

    • .merit_func - array including all merit function values

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

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

    • .Status - reason of termination

InitialDuplo(options)[source]

Function for initializing the parameters of DuploBacktrack, DuploConStep, and DuploDoubBacktrack. If some parameters specified by the user InitialDuplo uses these parameters. Otherwise, the default values will be used.

USAGE

[MaxNumIter, MaxNumMapEval, TimeLimit, epsilon, alpha, beta, sigma, l, tauBar, lambda_min, lambda_max, flag_x_error, flag_psi_error, flag_time, Stopping_Crit] = InitialDuplo (options)

INPUTS

options – structure including the parameteres of scheme

  • .MaxNumIter - maximum number of iterations

  • .MaxNumMapEval - maximum number of function evaluations

  • .TimeLimit - maximum running time

  • .epsilon - accuracy parameter

  • .x_opt - optimizer

  • .psi_opt - optimum

  • .alpha - constant with \(\alpha < 2 \sigma\)

  • .beta - reduction constant of the line search

  • .sigma - strong duplomonotone parameter

  • .l - Lipschitz continuity constant of f

  • .tauBar - a constant for determining the step-size

  • .lambda_min - lower bound of the step-size

  • .lambda_max - upper bound of the step-size

  • .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 \(||nfxk|| \leq \epsilon\)

    2. stop if MaxNumIter is reached

    3. stop if MaxNumMapEval is reached

    4. stop if TimeLimit is reached

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

OUTPUTS

  • MaxNumIter – maximum number of iterations

  • MaxNumMapEval – maximum number of function evaluations

  • TimeLimit – maximum running time

  • epsilon – accuracy parameter

  • x_opt – optimizer

  • psi_opt – optimum

  • alpha – a constant

  • beta – a constant

  • 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

Rate_function(opt, varargin)[source]
USAGE

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

    • .k - initial kinetic

OUTPUT

fxk – the vector f(xk)

StopCritDuplo(nfxk, Niter, Nmap, T, MaxNumIter, MaxNumMapEval, TimeLimit, epsilon, Stopping_Crit)[source]

Function checking that one of the stopping criteria holds to terminate LLM and GLM. It perepares the status determining why the algorithm is stopped.

USAGE

[StopFlag, Status] = StopCritDuplo (nfxk,Niter, Nmap, T, MaxNumIter, MaxNumMapEval, TimeLimit, epsilon, Stopping_Crit)

INPUTS

  • nhxk – the norm 2 of h(xk)

  • Niter – the number of iterations

  • Nmap – the number of mapping calls

  • T – the running time

  • MaxNumIter – maximum number of iterations

  • MaxNumMapEval – maximum number of function evaluations

  • TimeLimit – maximum running time

  • epsilon – accuracy parameter

  • Stopping_Crit – stopping criterion:

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

    2. stop if MaxNumIter is reached

    3. stop if MaxNumMapEval is reached

    4. stop if TimeLimit is reached

    5. 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