Geodesic
Ersatz function method for minimizing a finite-valued function on a compact set
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 2, pp. 195-207 
A. I. Ryabikov. Ersatz function method for minimizing a finite-valued function on a compact set. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 2, pp. 195-207. http://geodesic.mathdoc.fr/item/ZVMMF_2014_54_2_a2/
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     title = {Ersatz function method for minimizing a finite-valued function on a compact set},
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Voir la notice de l'article provenant de la source Math-Net.Ru

A method is proposed for solving optimization problems with continuous variables and a function taking a large finite set of values. Problems of this type arise in the multicriteria construction of a control rule for a discrete-time dynamical system whose performance criteria coincide with the number of violations of requirements imposed on the system. The rule depends on a finite set of parameters whose set of admissible values defines a collection of admissible control rules. An example is the problem of choosing a control rule for a cascade of reservoirs. The optimization method is based on solving a modified problem in which the original function is replaced by a continuous ersatz function. A theorem on the relation between the average-minimal values of the original and ersatz functions is proved. Optimization problems are solved with power-law ersatz functions, and the influence exerted by the exponent on the quality of the solution is determined. It is experimentally shown that the solutions produced by the method are of fairly high quality.

[1] Lotov A. V., Riabikov A. I., “Multi-criteria design of control rules for nonlinear dynamic problems by using visualization of the Pareto frontier”, Abstracts of the 2$^\mathrm{nd}$ Int. Conf. Optimizat. Appl., Optima-2011, Izd-vo VTs RAN, M., 2011, 157–159

[2] Lotov A. V., Pospelova I. I., Mnogokriterialnye zadachi prinyatiya reshenii, Izd-vo Maks Press, M., 2008

[3] Evtushenko Yu. G., Posypkin M. A., “Primenenie metoda neravnomernykh pokrytii dlya globalnoi optimizatsii chastichno tselochislennykh nelineinykh zadach”, Zh. vychisl. matem. i matem. fiz., 51:8 (2011), 1376–1389 | MR | Zbl

[4] Buber A. L., “Operativnoe upravlenie vodnymi resursami vodokhranilisch Angaro-Eniseiskogo kaskada GES: bezopasnost, podderzhka prinyatiya reshenii, optimalnoe upravlenie”, Rol melioratsii v obespechenii prodovolstvennoi i ekologicheskoi bezopasnosti Rossii, Materialy Mezhdunar. nauch.-prakt. konf., v. 2, M., 2009, 26–32

[5] Evtushenko Yu. G., Metody resheniya ekstremalnykh zadach i ikh primenenie v sistemakh optimizatsii, Nauka, M., 1982 | MR | Zbl

[6] Zhiglyavskii A. A., Zhilinskas A. G., Metody poiska globalnogo ekstremuma, Nauka, M., 1991 | MR

[7] Kamenev G. K., “Approksimatsiya vpolne ogranichennykh mnozhestv metodom glubokikh yam”, Zh. vychisl. matem. i matem. fiz., 41:11 (2001), 1751–1760 | MR | Zbl

[8] Kamenev G. K., “Issledovanie adaptivnogo odnofaznogo metoda approksimatsii mnogomernoi granitsy Pareto v nelineinykh sistemakh”, Zh. vychisl. matem. i matem. fiz., 49:12 (2009), 2103–2113 | MR