Combination of lexicography and nonlinear programming methods for multiextreme solutions searching

Аuthors

Petrovskii V. S.

Military industrial corporation «NPO Mashinostroyenia», 33, Gagarina str., Reutov, Moscow region, 143966, Russia

e-mail: desnavp2@mail.ru

Abstract

One of progressive applied methods for solution to a problem of nonlinear programming in difficult technical system such as the aircraft is presented. Process of a system shape choice always takes place in conditions of incomplete information. It is caused by uncertainty both in real conditions application and prediction of preliminary technical solutions feasibility. The choice of the only optimum decision is conditional and therefore unacceptable. With respect to conditions the one decision can be difficult to realize and very expensive in comparison with other possible solutions, which earlier were seemed not optimum and therefore were not revealed by traditional methods.
To identify the set of possible extreme variants, which at further specification of conditions could become an optimum (more precisely to tell, the rational supremum by principle), it is proposed to combine the method of lexicographic multicriteria optimization with searching method with an adjustable choice for direction and size of searching steps and with fixing for partial extrema. The lexicographic component of [1] method allows to cope with difficulties of multicriteria optimization when each purpose of priority criterion leads to the alternative solutions. Uncertainty is parried by consecutive purpose of a priority row.
The problem of a choice for system design shape is solved as follows:

• some options for lexicographic sequence of criteria are formed by their priority;
• rational decisions versions of system shape for each lexicographic sequence of criteria are defined;
• the common or close shape decisions for all options of lexicographic sequence of criteria are shown.

The stated approach prepares a basis for repeated optimization.
For the repeated solution to an optimization problem the one of progressive applied methods of the solution to a problem of nonlinear programming, so-called gradient methods of a big step [2, 3], is offered. Feature of the method is that the choice of the next step direction of search is carried out with respect to the maximum gradient of function change. In the chosen direction the all partial extrema are fixed and the size of the next searching step is formed. The method allows to significantly reduce searching time in comparison with well-known methods of «formal search» and increase a number of settlement options.
Thus, the advantage of the combined method allows to optimize function with a large number of variables and many extrema with linear and nonlinear restrictions in searching area. Such opportunity is especially actual for solution to engineering tasks in choice of difficult technical shape or other (information, economic, social, etc.) systems.

Keywords:

choice of difficult system shape, uncertainty of predicted conditions, lexicographic multicriteria optimization, multiextreme solutions, gradient method for extrema searching, searching area, linear restrictions, nonlinear restrictions

References

1. Podinovskii V.V. Leksikograficheskie zadachi optimizatsii (Lexicographic problems of optimization), Moscow, Voennaya akademiya im. F.E. Dzerzhinskogo, 1972, 80 p.
2. Zontendeik G. Metody vozmozhnykh napravlenii (Methods of the possible directions), Moscow, Nauka, 1971, 170 p.
3. Grushchanskii V.A. Metody sravneniya effektivnosti adaptivnykh proektiruemykh letatelnykh apparatov (Comparison methods for adaptive designed aircraft efficiency), Moscow, MAI, 1990, 56 p.