Confidence Level Routine

Although the general, mass conservation, and semi-diffusive approaches can quickly provide confirmation of bistability for most examples, this may not always be the case. In fact, an important item of discussion is that these approaches cannot exclude bistability, even if a large amount of random decision vectors are explored. It is this uncertainty that we wish to address. This is done by assigning a probability that the minimum objective function value achieved is equal to the true global minimum. We achieve this probability by considering a slightly modified version of the unified Bayesian stopping rule in [BGS04] and Theorem 4.1 of [SF87], where the rule was first established.

Let \alpha_k and \alpha^* denote the probability that the optimization routine has converged to the local minimum objective function value, say f_k, and global minimum objective function value, say f^*. Assuming that \alpha^* \geq \alpha_k for all local minimum values f_k we may then state that the probability that \tilde{f} = f^* is as follows:

Pr[\tilde{f} = f^*] \geq q(n, r) = 1 - \dfrac{(n + a + b - 1)! (2n + b - r - 1)!}{(2n + a + b - 1)! (n + b-r-1)!},

where n is the number of initial decision vectors that are considered, \tilde{f} = min \{f_1, \dots, f_n \} , a and b are parameters of the Beta distribution \beta(a, b), and q(n, r) is the confidence level. We then let r be the number of f_k for k = 1, \dots, n that are in the neighborhood of \tilde{f}.

Given our minimum objective function value is zero, for some networks it may be the case that the f_k are nearly zero with respect to machine precision. For this reason, we say that f_k is in the neighborhood of \tilde{f} if

\dfrac{| \tilde{f} - f_k |}{\tilde{f}} \leq 10^{-2}.

This means that f_k is in the neighborhood of \tilde{f} if the relative error of f_k and \tilde{f} is less than 1%. If \tilde{f} is considered zero with respect to the system’s minimum positive normalized float, then we consider this value zero and provide q(n,r) = 1.0, skipping the computation of q(n,r). Thus, we can state that the probability that the obtained \tilde{f} is the global minimum (for the prescribed bounds of the decision vector) is greater than or equal to the confidence level q(n, r). Using the standard practice in statistics, it should be noted that q(n,r) \geq 0.95 is often considered an acceptable confidence level to make the conclusion that \tilde{f} is the global minimum of the objective function.

For information on how to enable the construction of a confidence level for each of the approaches, please refer to the following for each approach:

  • Mass conservation approach:
    • If using crnt4sbml.MassConservationApproach.run_optimization() set confidence_level_flag = True and and prescribe a value to change_in_rel_error (if applicable)
    • If using crnt4sbml.MassConservationApproach.run_mpi_optimization() set confidence_level_flag = True and and prescribe a value to change_in_rel_error (if applicable)
  • Semi-diffusive approach:
    • If using crnt4sbml.SemiDiffusiveApproach.run_optimization() set confidence_level_flag = True and prescribe a value to change_in_rel_error (if applicable)
    • If using crnt4sbml.SemiDiffusiveApproach.run_mpi_optimization() set confidence_level_flag = True and and prescribe a value to change_in_rel_error (if applicable)
  • General approach: