Numerical Optimization Routine

In [OMYS17] it is suggested to use the ESS algorithm in the MEIGO toolbox to solve the constrained global optimization problem. Although evolutionary algorithms such as ESS can perform very well, they often need to be coupled with multi-start procedures to produce sensible results for complex reaction networks. In addition to this, to use the MEIGO toolbox within Python, a Python interface to R is required. This is not desirable, and for this reason we have constructed our own multi-start routine that compares favorably with the ESS routine for a general class of reaction networks.

The optimization routine utilizes two steps to achieve a minimization of the objective function:

  1. Multi-level feasible point method
  2. Hybrid global-local searches beginning at the feasibility points

Feasible Point Method

Both the mass conservation and semi-diffusive approach have constraints on the decision vector provided. These extra constraints coupled with the global optimization problem are difficult to solve and can often require many multi-starts to find a solution. This is due to the fact that multi-start routines often start at randomly generated values pulled from a uniform distribution, which do not satisfy the constraints. One way to begin the multi-start procedure in favorable positions is to generate starting points that already satisfy the constraints of the problem. We do this by conducting a feasible point method.

The feasible point method attempts to minimize the following objective function

f(\textbf{x}) = \sum_{i = 1}^I [v(g_i(\textbf{x}))]^2 + \sum_{j=1}^J [v(\textbf{x}_j)]^2 .

where v( \cdot) are violation functions for the constraint equations, g_i( \cdot), and variable bounds, \textbf{x}_j. The violation functions are defined as follows

Constraint Type Violation Function
g_i(\textbf{x}) \leq b max(0,g_i(\textbf{x})-b )
g_i(\textbf{x}) \geq b max(0,b - g_i(\textbf{x}))
g_i(\textbf{x}) = b | g_i(\textbf{x}) - b |
Variable Bounds Violation Function
\textbf{x}_j \leq b max(0,\textbf{x}_j -b )
\textbf{x}_j \geq b max(0,b - \textbf{x}_j)
\textbf{x}_j = b | \textbf{x}_j - b |

this is called a penalty method and is outlined in Chapter 18 of [Chi14]. For the mass conservation approach the constraint equations are defined as c_i \geq 0, where c_i are the species’ concentration expressions derived from the equilibrium manifold. The variable bounds for this approach are then defined by the bounds established for the decision vector. For the semi-diffusive approach the constraint equations are defined as \bar{p}_i(\mu) \geq 0 if c_i is a key species. Note that the constraint of \bar{p}_i(\mu) = 0 if c_i is not a key species is not considered in the optimization directly as they are satisfied by direct substitution. The variable bounds are again the bounds established for the decision vector. Notice that in both approaches we do not consider the rank constraints. In practice these are very difficult to satisfy via direct optimization. However, if the objective function is minimized, then the rank constraints have a very high likelihood of being satisfied.

Once the penalty function f(\cdot) is constructed we can then continue by minimizing it. We do this by conducting a multi-level multi-start method. First we generate a user defined amount of decision vectors using a random uniform distribution and then put them in the user defined bounds. Next, we minimize f(\cdot) using SciPy’s SLSQP function with a tolerance of 1e-16. Although it is often sufficient to just run SLSQP, in some cases if a minimum of zero is not achieved by this run, it is beneficial to also perform a minimization using Nelder-Mead starting from the minimum point found by SLSQP. To reduce runtimes, we do not run the Nelder-Mead routine if SLSQP returns an objective function value that is sufficiently small.

Hybrid Global-Local Searches

Using those decision vectors produced by the feasible point method, we now address the global optimization problem. For the mass conservation approach we let the objective function be:

F(\textbf{x}) = det(G)^2 + f(\textbf{x}),

and for the semi-diffusive approach we let the objective function be:

F(\textbf{x}) = det(S_{to}diag(\mu)Y_r^T)^2 + f(\textbf{x}),

where f(\textbf{x}) is the objective function formed by the feasible point method. Using the decision vectors produced by the feasible point method as starting points, we then run SciPy’s global optimization algorithm Basin-hopping. In addition to running this global optimization, we employ Nelder-Mead as a local minimizer. If the local minimizer returns an objective function value smaller than a user defined value of sys_min_val, then the result solution array from the minimizer is saved and returned to the user.

Pseudocode for Optimization Method

Establish bounds for decision vector.

Randomly generate niter parameter sets of decision vectors within the given bounds, say samples.

for i = 1 to niter

Let samples_i be a starting point for the feasible point method where f(\textbf{x}) is the objective function

if samples_i provides f(\textbf{x}) \leq machine epsilon

Run hybrid global-local search for F(\textbf{x}) objective function with \textbf{x} as starting point, providing \textbf{x}_{best}.

Store \textbf{x}_{best} and function values that are smaller than sys_min_val

Throw away samples_i