varKin

optimizeVKmodel(model, solver, x0, parms)[source]

Function for finding a solution of the nonlinear system \(h(x) = f(x) = 0\), x in \(R^m\), (I) or \(h(x) = (f(x)^T, l(x)^T)^T = 0\), x in \(R^m\), (II) where \(f(x) = [F - R, R - F]*exp(k + [F, R]^T * x)\), \(l(x) = L*exp(x) - l_0\), using the nonlinear unconstrained minimization \(\textrm{min}\ psi(x) = 1/2 ||h(x)||^2\) s.t. x in \(R^m\). with several solvers. For (I), one can use all the following solvers; however, for (II), one can only apply local solvers LLM_F, LLM_FY, LLM_YF, LLM and global solvers GLM_FY, GLM_YF, LevMar, LMLS, and LMTR. If sonver field is empty, the code will use LMTR as the default.

Usage

output = optimizeVKmodel(model, solver, x0, parms)

Inputs

  • model – stracture includes F, R and/or L
    • .F - forward stoichiometric matrix
    • .R - reverse stoichiometric matrix
    • .L - basis for the left null-space of N = R - F
    • .l0 - constant \(l_0\)
  • solver –
    • Local Levenberg-Marquardt (LM) solvers:
      • LLM - LM of Ahookhosh et al.
      • LLM_F - LM of Fischer
      • LLM_FY - LM of Fun and Yuan
      • LLM_YF - LM of Yamashita and Fukushima
    • Global Levenberg-Marquardt (LM) solvers:
      • LM_FY - LM of Fun and Yuan
      • LM_YF - LM of Yamashita and Fukushima
      • LevMar - LM of Kelly
      • LMLS - LM line search of Ahookhosh et al.
      • LMTR - LM trust-region of Ahookhosh et al.
    • DC programming solvers:
      • DCA - DC programming algorithm
      • BDCA - boosted DC programming algorithm
    • Duplomonotone derivative-free solvers:
      • BDF - Derivative-free duplomonotone method 1
      • CSDF - Derivative-free duplomonotone method 2
      • DBDF - Derivative-free duplomonotone method 3
  • Parameters for LLM_F, LLM_FY, LLM_YF, LLM, GLM_FY, GLM_YF, LevMar, LMLS, LMTR
  • parms – structure including the parameteres of schemes
    • .eta - constant for Levenberg-Marquardt parameter
    • .MaxNumIter - maximum number of iterations
    • .MaxNumMapEval - maximum number of function evaluations
    • .MaxNumGmapEval - maximum number of gradient evaluations
    • .TimeLimit - maximum running time
    • .epsilon - accuracy parameter
    • .x_opt - optimizer
    • .psi_opt - optimum
    • .adaptive - update lambda adaptively
    • .kin - kinetic parameter in \(R^{2n}\)
    • .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 \(||grad|| \leq \textrm{max}(\epsilon, \epsilon ^2 * ngradx_0)\)
      8. stop if \(||nhxk|| \leq \textrm{max}(\epsilon, \epsilon^2 * nhx_0)\)
      9. stop if (default) \(||hxk|| \leq \epsilon\) or MaxNumIter is reached
  • Parameters for DCA, BDCA
  • parms – structure including the parameteres of schemes
    • 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 search
    • .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
  • Parameters for BDF, CSDF, DBDF
  • parms – structure including the parameteres of schemes
    • 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 - is the backtarcking constant;
    • .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

Output

  • output – structure including more output information:
    • .x_best - the best approximation of the optimizer
    • .psi_best - the best approximation of the optimum
    • .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 \(norm(x_k(:)-x_{opt}(:))/norm(x_{opt})\)
    • .psi_error - relative error \((psi_k-psi_{opt})/(psi_0-psi_{opt}))\)
    • .Status - reason of termination
    • .Time - running time of all iterations