Bdcamethods

BDCA(mapp, x0, options)[source]

BDCA 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] = BDCA (mapp, x0, options)

INPUTS:

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

  • x0 – initial point

  • options – structure including the parameteres of scheme

    • .F - forward stoichiometric matrix

    • .R - reverse stoichiometric matrix

    • .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 for the line searche

    • .beta - backtarcking constant

    • .lambda_bar - starting step-size for the line search

    • .rho - strong convexity parameter

    • .kin - kinetic parameter in R^(2n)

    • .flag_line_search - “Armijo” or “Quadratic_interpolation”

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

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

    • .Status - reason of termination

DCA(mapp, x0, options)[source]

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

USAGE:

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

INPUTS:

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

  • x0 – initial point

  • options – structure including the parameteres of scheme

    • .F - forward stoichiometric matrix

    • .R - reverse stoichiometric matrix

    • .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 for the line searche

    • .beta - backtarcking constant

    • .lambda_bar - starting step-size for the line search

    • .rho - strong convexity parameter

    • .kin - kinetic parameter in \(R^{2n}\)

    • .flag_line_search - “Armijo” or “Quadratic_interpolation”

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

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

    • .Status - reason of termination

FuncGradHessSub(x, y, F, R, kin, rho)[source]

SubGradHess provides the function value, the gradient, and the Hessian of the subproblem of DCA and BDCA to be used in fminunc.

USAGE:

[func, grad, Hess] = FuncGradHessSub (x, y, F, R, kin, rho)

INPUTS:

  • x – point

  • y – point

  • F – Forward stoichiometric matrix

  • R – Reverse stoichiometric matrix

  • kin – kinetics parameter in \(R^{2n}\)

  • rho – strongly comvex modulus

OUTPUTS:

  • func – function value

  • grad – gradient

  • Hess – Hessian

InitialBDCA(options)[source]

InitialBDCA is a function for initializing the parameters of BDCA and DCA. 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, lambda_bar, rho, kin, flag_line_search, flag_x_error, flag_psi_error, flag_time, Stopping_Crit] = InitialBDCA (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 for the line searche

  • .beta - backtarcking constant

  • .lambda_bar - starting step-size for the line search

  • .rho - strong convexity parameter

  • .kin - kinetic parameter in \(R^{2n}\)

  • .flag_line_search - “Armijo” or “Quadratic_interpolation”

  • .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

  • alpha – constant for the line search

  • beta – backtracking constant

  • lambda_bar – starting step-size for the line search

  • rho – strong convexity parameter

  • kin – kinetic parameter in \(R^{2n}\)

  • flag_line_search – “Armijo” or “Quadratic_interpolation”

  • 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]

Computes the rate function h(xk) for the current point, given the forward/reverse stoichiometric matrices and kinetic parameter in opt

USAGE:

fxk = Rate_function (opt, varargin)

INPUTS:

  • xk – current point;

  • opt – structure includes required parameters:

    • .FR - concatenation of forward and reverse stoichiometric matrix

    • .FR_RF - FR - RF, difference of forward-reverse and reverse-forward concatenations

    • .k - initial kinetic

OUTPUT:

fxk – the vector h(xk)

StopCritBDCA(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 perepare the status determining why the algorithm is stopped.

USAGE:

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

INPUTS:

  • nfxk – 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