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Verified methods for computing Pareto sets: General algorithmic analysis

G.-Tóth B., Kreinovich V.

International Journal of Applied Mathematics and Computer Science

2009

Vol. 19, no 3

369-380

In many engineering problems, we face multi-objective optimization, with several objective functions f1, . . . , fn. We want to provide the user with the Pareto set-a set of all possible solutions x which cannot be improved in all categories (i.e., for which fj (x') fj (x) for all j and fj(x') > fj(x) for some j is impossible). The user should be able to select an appropriate trade-off between, say, cost and durability. We extend the general results about (verified) algorithmic computability of maxima locations to show that Pareto sets can also be computed.

Interval mathematics for analysis of multi-level granularity

Kreinovich V., Alo R.

Archives of Control Sciences

2002

Vol. 12, no. 4

323-350

The more complex the problem, the more complex the system necessary for solving this problem. For very complex problems, it is no longer possible to design the corresponding system on a single resolution level, it becomes necessary to have multi-level, multiresolutional systems, with multi-level granulation. When analyzing such systems - e.g., when estimating their performance and/or their intelligence - it is reasonable to use the multi-level character of these systems: first, we analyze the system on the low-resolution level, and then we sharpen the results of the low-resolution analysis by considering higher-resolution representations of the analyzed system. The analysis of the low-resolution level provides us with an approximate value of the desired performance characteristic. In order to make a definite conclusion, we need to know the accuracy of this approximation. In this paper, we describe interval mathematics- a methodology for estimating such accuracy. The resulting interval approach is also extremely important for tessellating the space of search when searching for optimal control. We overview the corresponding theoretical results, and present several case studies.