zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

92B05General biology and biomathematics
65K10Optimization techniques (numerical methods)
93B30System identification
49N70Differential games in calculus of variations
49N75Pursuit and evasion games in calculus of variations
Full Text: DOI
[1] Cherruault, Y.: Biomathématiques. Que sais-je? (1983)
[2] Kolmogorov, A. N.: On the representation of continuous functions of several variables by superpositions of continuous functions of one variable and addition. Dokl. akad. Nauk. 114, 679-681 (1957) · Zbl 0090.27103
[3] Vaĭndiner, A. L.: Approximation of continuous and differentiable functions of several variables by generalized polynomials (Finite linear combinations of functions of fewer variables). Dokl. akad. Nauk 192, No. 3, 648-652 (1970) · Zbl 0215.46501
[4] Cherruault, Y.: A new method for global optimization. Kybernetes 19, No. 3, 19-32 (1990) · Zbl 0701.90083
[5] Cherruault, Y.: Mathematical modelling in biomedicine, optimal control of biomedical systems. (1986)
[6] Cherruault, Y.; Guillez, A.: Algorithmes originaux pour la simulation numérique et l’optimisation. Industries alimentaires et agricoles (IAS) 10, 879-888 (1987)
[7] Y. Cherruault, New deterministic methods for global optimization and applications to Biomedicine, Int. Journal of Bio-medical Computing (to appear)
[8] Cherruault, Y.: Modélisation et méthodes mathématiques en biomédicine. (1977)
[9] Horst, R.: Deterministic methods in constrained global optimization, some recent advances and new fields of applications. Naval research logistics 37, 433-471 (1990) · Zbl 0709.90093
[10] Törn, A.; Z\breve{}ilinskas, A.: Global optimization. (1987)
[11] Schwartz, L.: Etude des sommes d’exponentielles. (1959) · Zbl 0092.06302
[12] Bremermann, H.: A method of unconstrained global optimization. Math. biosc. 9, 1-15 (1970) · Zbl 0212.51204
[13] Horst, R.; Tuy, H.: Global optimization, deterministic approaches. (1990) · Zbl 0704.90057
[14] Van Laarhoven, P. J. M.; Aarts, E. H. L.: Simulated annealing: theory and applications. (1987) · Zbl 0643.65028
[15] Spang, H. A.: A review of minimization techniques for nonlinear functions. SIAM review 4, No. 4 (1962) · Zbl 0112.12205
[16] Henrici, P.: Discrete variable methods in ordinary differential equations. (1964) · Zbl 0112.34901