Multi-Objective Optimization Problem: Stability against Perturbations of Input Data in Vector-Valued Criterion
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MULTI-OBJECTIVE OPTIMIZATION PROBLEM: STABILITY AGAINST PERTURBATIONS OF INPUT DATA IN VECTOR-VALUED CRITERION Ò. Ò. Lebedeva,1† N. V. Semenova,1‡ and Ò. ². Sergienko1††
UDC 519.8
Abstract. The conditions of stability against input data perturbations in vector-valued criterion for multi-objective optimization problem with continuous partial criterion functions and feasible set of arbitrary structure are established. The sufficient and necessary conditions of three types of stability for the problem of finding Pareto optimal solutions are proved. Keywords: vector optimization problem, vector-valued criterion, stability, Pareto optimal solutions, Slater set, Smale set, perturbations of input data. INTRODUCTION The paper pertains to the theoretical field of studies in the problem of stability of multi-objective (vector) optimization problems. This field is related to finding and analyzing the conditions whereby a set of Pareto, Slater or Smale optimal solutions of a problem possesses some predetermined property that characterizes in a certain way its stability against small perturbations of input data. The paper continues studies in the correctness of vector optimization problems, including their solvability and stability, presented in particular in [1–9]. The results described therein expand the well-known class of vector optimization problems, stable with respect to input data perturbations for vector-valued criterion. Another well-known field in the analysis of the stability problem is oriented toward obtaining and investigating the quantitative characteristics of feasible modifications in input data of the problem, in particular, radius of the maximum stability sphere (see, for example, [10–12]). PROBLEM STATEMENT. BASIC DEFINITIONS Let us consider a vector optimization problem of the form
Q ( F , X ): max{F ( x ) | x Î X } ,
(1)
where X is a set from R n of arbitrary structure, probably discrete; R n is an n-dimensional real space; F ( x ) = ( f 1 ( x ), K , f l ( x )), l ³ 2 , f i : R n ® R 1 is a continuous function, i Î N l = {1, K , l } , and X ¹ Æ . Let the problem (1) be: find elements of the set of Pareto optimal solutions
P ( F , X ) = {x Î X | p ( x, F , X ) = Æ}, where p( x, F , X ) = { y Î X | F ( y) ³ F ( x ), F ( y) ¹ F ( x )}. 1
V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]; ‡[email protected]; ††[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 6, November–December, 2020, pp. 107–114. Original article submitted August 17, 2020.
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1060-0396/20/5606-0953 ©2020 Springer Science+Business Media, LLC
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Let us also consider the sets of Slater optimal solutions:
Sl ( F , X ) = {x Î X | s ( x, F , X ) = Æ}, where s( x, F , X ) = { y Î X | F ( y) > F ( x )} , and of Smale optimal ones:
Sm( F , X ) = {x Î X | h ( x, F , X ) = Æ} , where h( x, F , X ) = { y Î X | y ¹ x, F ( y) ³ F ( x )}. It can be easily seen that Sm( F , X ) Ì P ( F , X ) Ì Sl( F , X )
(2)
and " x Î X s ( x, F , X ) Ì p ( x, F ,
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