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
    • .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
    • .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,y – points
  • F – Forward stoichiometric matrix
  • kin – kinetics parameter in \(R^{2n}\)
  • rho – strongly comvex modulus

Outputs

  • f – function value
  • grad – gradient
  • H – 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.

Input

  • 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
  • x_opt – optimizer
  • psi_opt – optimum
  • alpha – constant for the line search
  • beta – backtracking constant
  • lambda_bar – starting step-size for 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
Rate_function(opt, varargin)[source]

Usage

fxk = 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
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

  • 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