Abstract
The use of Helmholtz (diffraction and diffusion) and heat conductivity equations for global optimization is described in Part I. Here numerical realization of these ideas is studied and the superlinear rate of convergence is demonstrated. Numerical methods are based on the idea that the solutions of diffraction (diffusion) equations ϕ(x,ω) and heat conductivity equation U(x,t) are convex and concave functions, respectively, in the neighborhood of the point of global minimum in some modified aim function for which the point of global minimum does not change. An idea for the construction of iterative algorithms is developed, in which a programmer, using computation results, actively participates in computations, instructing the computer as to which domains and distribution densities of random vectors are to be used.
Similar content being viewed by others
REFERENCES
Kaplinskii, A.I., Pesin, A.M., and Propoi, A.I., An Investigation into Optimization Search Methods based on Potential Theory. I, Avtom.Telemekh., 1994, no. 9, pp. 97–105.
Kaplinskii, A.I., Pesin, A.M., and Propoi, A.I., An Investigation into Optimization Search Methods based on Potential Theory. II, Avtom.Telemekh., 1994, no. 10, pp. 67–73.
Kaplinskii, A.I., Pesin, A.M., and Propoi, A.I., An Investigation into Optimization Search Methods based on Potential Theory. III, Avtom.Telemekh., 1994, no. 11, pp. 66–73.
Kaplinskii, A.I. and Propoi, A.I., First-Order Nonlocal Optimization Methods based on Potential Theory, Avtom.Telemekh., 1994, no. 7, pp. 94–103.
Kaplinskii, A.I. and Propoi, A.I., Second-Order Refinement Conditions based on Potential Theory, Avtom.Telemekh., 1994, no. 8, pp. 104–113.
Prudnikov, I.M., A Global Optimization Method and Estimation of Its Convergence Rate, Avtom.Telemekh., 1993, no. 12, pp. 72–81.
Kaplinskii, A.I. and Propoi, A.I., Design of Nonlocal Optimization Algorithms: A Variational Approach, Preprint of Inst.of Syst.Res., 1986.
Prudnikov, I.M., Application of Potential Methods in Optimization of a Function on a Set Defined as a System of Inequalities, Avtom.Telemekh., 1996, no. 2, pp. 66–82.
Clark, F.H., Optimization and Nonsmooth Analysis, New York: Wiley, 1983. Translated under the title Optimizatsiya i negladkii analiz, Moscow: Nauka, 1984.
Dem'yanov, V.F. and Rubinov, A.M., Osnovy negladkogo analiza i kvazidifferentsial'noe ischislenie (Elements of Nonsmooth Analysis and Quasidifferential Calculus), Moscow: Nauka, 1990.
Prudnikov, I.M., Lower Convex Approximations for Lipschitz Functions, Zh.Vychisl.Mat.Mat.Fiz., 2000, vol. 40, no. 3, pp. 3780–386.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Prudnikov, I.M. The Use of Certain Equations of Mathematical Physics in Optimization of a Function on a Set. II. Automation and Remote Control 63, 1891–1899 (2002). https://doi.org/10.1023/A:1021631112889
Issue Date:
DOI: https://doi.org/10.1023/A:1021631112889