THE OPTIMIZATION IN CONDITION OF UNCERTAINTY BY DETERMINATION METHOD

V. I. Levin

Abstract


The existing approaches to the optimization (optimal design) of systems under uncertainty are considered. An exact formulation of
problem of constrained optimization under interval uncertainty of the parameters of the objective function and constraints is given. In this
connection the mathematical theory of comparison of intervals is set out, including a precise definition of the maximal and minimal intervals,
conditions for existence of such intervals and algorithms for finding them. Idea of solving constrained optimization problems under interval
uncertainty of its parameters is proposed. This idea is based on the rules of the mathematical theory of comparison of intervals which allows
replace the comparison of intervals and determination of maximal and minimal interval by comparing their lower and upper bounds. On basis
of the proposed idea the determination method which allows solve the problem of constrained optimization under interval uncertainty parameters by reducing it to two entirely certain optimization problems of the same type is formulated and proved. We formulate and prove
a theorem that defines the solution of the problem of constrained optimization under interval uncertainty of parameters through solutions of two fully certain optimization problems. Also the theorem that defines the necessary and sufficient condition for existence of a solution of
constraint optimization under interval uncertainty is formulated and proved. The algorithm of solving constrained optimization under interval
uncertainty parameters that implements a method of determination is constructed and consists of 4 steps. The example of the algorithm is
given. The interval assignment task is selected as a problem to be solved is selected. A comparison of our approach to solving constrained
optimization problems with incompletely defined parameters with other methods for solving such problems (deterministic, probabilistic and
fuzzy) is done. Advantages and disadvantages of different methods are listed. It is emphasized that the proposed in the article approach allows
us to reduce the optimization of incompletely specified functions to fully optimize certain functions strictly mathematically rather than
heuristically, as is done in well-known approaches.

Keywords


optimization, uncertainty, optimization with interval uncertainty, determination.

References


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DOI: https://doi.org/10.15588/1607-3274-2015-4-15



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