Difference between revisions of "Optimization problem"
From BioUML platform
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<li>deterministic method of global optimization glbSolve [4]; | <li>deterministic method of global optimization glbSolve [4]; | ||
<li>adaptive simulated annealing (ASA) [5]. | <li>adaptive simulated annealing (ASA) [5]. | ||
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| + | The below table shows the generic scheme of the optimization process for these methods. SRES, MOCell, PSO and glbSolve run a predefined number of iterations ''N<sub>it</sub>'' considering a sequence of sets (populations) ''P<sup>i</sup>'', ''i'' = 0,…,''N<sub>it</sub>'' − 1, of potential solutions (guesses). In the case of the first three methods, the size ''s'' ∈ ℕ<sup>+</sup> of the population is fixed, whereas in glbSolve the initial population ''P''<sup>0</sup> consists of one guess, while the size ''s''<sub>''k''+ 1</sub> of the population ''P''<sup>''k''+1</sup> is found during the iteration | ||
| + | with the number ''k'' = 0,…, ''N<sub>it</sub>'' − 1. The method ASA considers sequentially generated guesses ''x<sup>k</sup>'' ∈ Ω, ''k'' ∈ ℕ<sup>+</sup>, and stops if distance between ''x<sup>k</sup>'' and ''x''<sup>''k''+1</sup> defined as Euclidean norm | ||
| + | |||
| + | [[File:optimization_formula_4.png]] | ||
| + | |||
| + | becomes less than a predefined accuracy ε. | ||
==References== | ==References== | ||
Revision as of 16:43, 12 March 2019
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 ⊆ ℝN defined by a set of equality and/or inequality constraints on x. Since the equality gs(x) = 0 can be replaced by two inequalities gs(x) ≤ 0 and –gs(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:
becomes less than a predefined accuracy ε.
References
- Runarsson T.P., Yao X. Stochastic ranking for constrained evolutionary optimization. IEEE Transactions on Evolutionary Computation. 2000. 4(3):284–294.
- 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.
- 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.
- Björkman M., Holmström K. Global Optimization Using the DIRECT Algorithm in Matlab. Advanced Modeling and Optimization. 1999. 1(2):17–37.
- Ingber L. Adaptive simulated annealing (ASA): Lessons learned. Control and Cybernetics. 1996. 25(1):33–54.
