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Global optimization in biology and medicine. (English) Zbl 0808.92001
Global optimization techniques are fundamental for solving identification problems coming from modelling. They also play a great role for optimizing biological processes. But it is also possible to solve functional equations (partial differential, integral, etc.) by using a minimization technique with an error functional defined from experimental data and functional equations. It suffices to express the solution under a mathematical expression (polynomial or exponential development, spline approximation, etc.) and to identify the unknown parameters in the mathematical definition by minimizing the error functional. Thus, global optimization techniques are very precious and important in numerical mathematics. We proposed two kinds of methods (deterministic and stochastic) for solving global optimization problems. For the deterministic case, we presented three techniques: -- the first is to choose an approximation equal to the sum of functions depending only on a single variable. The minimization problem is brought back to the minimization of functions depending on a single variable; -- the second technique adds new variables and permits us to solve an optimization problem by solving a sequence of linear problems; -- the third method, called Alienor, is based on a reducing transformation allowing the approximation of $n$ variables by a single one. A minimization problem according to $n$ variables becomes an approximated minimization problem depending on one variable. For the stochastic case, we developed Monte Carlo methods. Two techniques were presented: the simulated annealing method; and the Bremermann method [{\it H. Bremermann}, Math. Biosci. 9, 1-15 (1970; Zbl 0212.512)]. Then, applications to identification problems and to process optimization were given. We tried to compare the complexity (calculations times) involved by these methods.

##### MSC:
 92B05 General biology and biomathematics 65K10 Optimization techniques (numerical methods) 93B30 System identification 49N70 Differential games in calculus of variations 49N75 Pursuit and evasion games in calculus of variations
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##### References:
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