Optimization problem

The general nonlinear optimization problem [1] can be formulated as follows: find a minimum of the objective function ϕ(x), where x lies in the intersection of the N-dimensional search space

and the admissible region ℱ ⊆ ℝ^N defined by a set of equality and/or inequality constraints on x. Since the equality g_s(x) = 0 can be replaced by two inequalities g_s(x) ≤ 0 and –g_s(x) ≤ 0, the admissible region can be defined without loss of generality as

In order to get solution situated inside ℱ, we minimize the penalty function

The problem could be solved by different optimization methods. We implemented the following of them in the BioUML software:

stochastic ranking evolution strategy (SRES) [1];

cellular genetic algorithm MOCell [2];

particle swarm optimization (PSO) [3];

deterministic method of global optimization glbSolve [4];

adaptive simulated annealing (ASA) [5].

References

1. Runarsson T.P., Yao X. Stochastic ranking for constrained evolutionary optimization. IEEE Transactions on Evolutionary Computation. 2000. 4(3):284–294.
2. Nebro A.J., Durillo J.J., Luna F., Dorronsoro B., Alba E. MOCell: A cellular genetic algorithm for multiobjective optimization. International Journal of Intelligent Systems. 2009. 24(7):726–746.
3. Sierra M.R., Coello C.A. Improving PSO-Based Multi-objective Optimization Using Crowding, Mutation and ∈-Dominance. Evolutionary Multi-Criterion Optimization. Lecture Notes in Computer Scienc. 2005. 3410:505-519.
4. Björkman M., Holmström K. Global Optimization Using the DIRECT Algorithm in Matlab. Advanced Modeling and Optimization. 1999. 1(2):17–37.
5. Ingber L. Adaptive simulated annealing (ASA): Lessons learned. Control and Cybernetics. 1996. 25(1):33–54.