Просмотр исходного текста страницы 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

[[File:optimization_formula_1.png]]

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

[[File:optimization_formula_2.png]]

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

[[File:optimization_formula_3.png]]

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

<ul>
<li>stochastic ranking evolution strategy (SRES) [1];</li>
<li>cellular genetic algorithm MOCell [2];</li>
<li>particle swarm optimization (PSO) [3];</li>
<li>deterministic method of global optimization glbSolve [4];</li>
<li>adaptive simulated annealing (ASA) [5].</li>
</ul>

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.

[[File:optimization_table_1.png]]

All methods, excepting glbSolve, are stochastic and seek global minimum of the function ϕ taking into account the admissible region ℱ. Thus, a guess ''x'' ∈ Ω is more preferable than a guess ''y'' ∈ Ω at some iteration of methods, if ψ(''x'') = 0 and ψ(''y'') ≠ 0 or ψ(''x'') < ψ(''y''). The method glbSolve is suited to solve only the problems with Ω ⊆ ℱ. Values of the function ψ are calculated but do not affect on the generation of potential solutions.

==Application of non-linear optimization to systems biology==

We assume that a mathematical model of some biological process consists of a set of chemical species ''S'' = {''S''<sub>1</sub>,…,''S<sub>m</sub>''} associated with variables ''C''(''t'') = (''C''<sub>1</sub>(''t''),…,''C<sub>m</sub>''(''t'')) representing their concentrations, and a set of biochemical reactions ℛ = {''R''<sub>1</sub>,…,''R<sub>n</sub>''} with rates ''v''(''t'') = (''v''<sub>1</sub>(''t''),…,''v<sub>n</sub>''(''t'')) depending on a set of kinetic constants ''K''. Reaction rates are modeled by standard laws of chemical kinetics. A Cauchy problem for ordinary differential equations representing a linear combination of reaction rates is used to describe the model behavior over time:

[[File:optimization_formula_5.png]]&emsp;&emsp;&emsp;'''(1)'''

Here ''N'' is a stoichiometric matrix of ''n'' by ''m''. We say that ''C<sup>ss</sup>'' is a steady state of the system (1) if

[[File:optimization_formula_6.png]]

==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.