Stability and unboundedness of vector optimization problems

Сергиенко, И. В.; Kozeratskaya, L. N.; Kononova, Anna

doi:10.1007/bf02665934

Cited by 6 publications

(6 citation statements)

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“…For the case where the value of the perturbation parameter t differs from unity, the result presented in Theorem 2 was specified. THEOREM 4 [9]. " Î = t…”

Section: Proofmentioning

confidence: 99%

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Properties of Perturbed Cones Ordering the Set of Feasible Solutions of Vector Optimization Problem1

2014

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The influence of perturbations of initial data on the solutions of multicriteria optimization problems is considered. The properties of perturbed cones, which partially order the feasible domain of the vector optimization problem with respect to linear objective functions are analyzed. The structure of the set of specific perturbed ordering cones with different values of the parameter of perturbations of the initial data is investigated.Keywords: vector optimization problems, perturbation of initial data, perturbed ordering cones.In the paper, we analyze how the uncertainty in initial data influences the solutions of vector optimization problems with many linear criteria. There are different sources of uncertainty: data variation in time, imperfect initial data, subjective data, incomplete data, measurement errors. In optimization problems, especially those where variables take discrete values, small perturbations in initial data can lead to solutions that greatly differ from the true ones. The results of solution of such problems are often unpredictable, even for minor variations in initial data. In this connection, it is important to develop a toolkit to predict the influence of perturbations in initial data on the results being obtained. The studies are aimed at expanding the capabilities of the use of convex cones, which order the sets of feasible solutions of vector optimization problems with respect to partial criteria, for the analysis of the influence of perturbations in initial data on the solutions of such problems.Consider a vector optimization problem

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“…For the case where the value of the perturbation parameter t differs from unity, the result presented in Theorem 2 was specified. THEOREM 4 [9]. " Î = t…”

Section: Proofmentioning

confidence: 99%

“…In the present paper, we will continue (from [5][6][7][8][9][10]) the analysis of properties of ordering cones perturbed specially according to formulas (3) and (4). Based on some of these properties, we will develop an approach [6,8] to the regularization of problem (1) with integer variables, which is possibly unstable against perturbations of initial data.…”

mentioning

confidence: 99%

Properties of Perturbed Cones Ordering the Set of Feasible Solutions of Vector Optimization Problem1

2014

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“…In the next section we represent particular perturbations, which are an extension of the work presented in [29], of the ordering cone K as a tool to analyze and predict changes in the solutions of multi-objective optimization problems with integer and/or continuous variables due to input data changes. In the fourth section we formulate and prove the properties of these cones, which can be used to:…”

Section: Example 1 Considermentioning

confidence: 99%

“…In this paper ordering cones are defined as K ={x ∈ R n : Cx 0}. Although perturbations of ordering cones were considered in [17,29,30], there was no comprehensive study of specific perturbations of ordering cones. Such a comprehensive study of specific cone perturbations is a central contribution of our work.…”

Section: Example 1 Considermentioning

confidence: 99%

“…Cones are an important construct in mathematics, have been widely studied [4,21,27], and have found application in various engineering and scientific problems [2,24]. The use of cones specifically for vector (multi-objective) optimization is discussed in [11,[28][29][30]. By definition [27], a subset K of R n is called a cone, if it is closed under positive scalar multiplication (i.e., λx ∈ K when x ∈ K and λ > 0).…”

Section: Example 1 Considermentioning

confidence: 99%

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Perturbed cones for analysis of uncertain multi-criteria optimization problems

Kozeratska

Forbes

Goebel

et al. 2004

Linear Algebra and its Applications

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Partial ordering of two quantities x and y (i.e., the ability to declare that x is better than y with respect to some decision criteria) can be stated mathematically as: x is better than y iff x − y ∈ K, where K is an ordering convex cone, not necessarily pointed. Cones can be very important in representing feasible domains (i.e., {Ax b} = M + G, where M is a bounded convex hull of a finite number of points and G is a convex cone). We consider specific perturbations of the Cone of Feasible Directions, which lead to a better feasible solution with respect to some decision criteria. Such cones are introduced as a tool to mitigate and analyze the effects of input data uncertainty on the solution of a given problem. Properties of this cone provide a basis to prove necessary and sufficient conditions for stable/unstable unboundedness of the multi-criteria optimization problem.

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Conditions of Pareto Optimization Problems Solvability: Stable and Unstable Solvability

Сергиенко

2017

Optimization Methods and Applications

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Stability and unboundedness of vector optimization problems

Cited by 6 publications

References 1 publication

Properties of Perturbed Cones Ordering the Set of Feasible Solutions of Vector Optimization Problem1

Properties of Perturbed Cones Ordering the Set of Feasible Solutions of Vector Optimization Problem1

Perturbed cones for analysis of uncertain multi-criteria optimization problems

Conditions of Pareto Optimization Problems Solvability: Stable and Unstable Solvability

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