Efficiency in uncertain variational control problems
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ORIGINAL ARTICLE
Efficiency in uncertain variational control problems Savin Treant¸a˘1 Received: 30 September 2019 / Accepted: 8 September 2020 Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract In this paper, considering the applications of interval analysis in various fields (such as artificial intelligence, neural computation, genetic algorithms, information theory or fuzzy logic), a new class of interval-valued variational control problems governed by multiple integral functionals, first-order PDE and inequality constraints is studied. More precisely, efficiency conditions for the considered uncertain variational control problem are formulated and proved. The sufficiency of Karush–Kuhn–Tucker conditions is established under some invexity and ðq; bÞ-quasiinvexity assumptions of the involved functionals. In addition, the paper is completed with illustrative applications (describing the controlled behavior of an artificial neural system) and the corresponding algorithm. Keywords LU-optimal solution Uncertain variational control problem Interval-valued variational problem ðq; bÞ-Quasiinvexity Genetic algorithms Mathematics Subject Classification 26B25 65K10 90C26 90C30 90C46 49K20 46T20 68T37
1 Introduction Because of the increasing complexity of the environment, the initial data often suffer from inaccuracy. In many practical situations, where data are only known to lie within intervals and only ranges of values are sought as satisfactory answers, the straightforward interval computation can yield the desired results. Examples of this type of application are the performance of artificial neural systems and certain chaotic phenomena, such as disasters and turbulences. Intervals and interval functions appear to be convenient means to describe the results of observations, which rarely yield exact real numbers and functions. In addition, many decisions must be made on the basis of a range of possible values of various parameters, and one is often interested in how a system will perform under a variety of conditions. Thus, by introducing interval functions and the corresponding analysis may lead to simpler
& Savin Treant¸ a˘ [email protected] 1
Department of Applied Mathematics, Faculty of Applied Sciences, University Politehnica of Bucharest, 313 Splaiul Independentei, 060042 Bucharest, Romania
models that will give satisfactory accuracy results for practical purposes (see, for instance [11]). In order to tackle the uncertainty in an optimization problem, the interval-valued optimization represents a new and growing branch of applied mathematics. The intervalvalued optimization problems may provide an alternative choice for considering the uncertainty into the optimization problems. Taking into account the applications of interval analysis in various fields, such as neural computation, artificial intelligence, genetic algorithms or fuzzy logic, the current paper is situated around the studies of uncertain optimization problem
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